text:
Incircle and excircles of a triangle
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"Incircle" redirects here. For incircles of non-triangle polygons, see Tangential quadrilateral and Tangential polygon.
A triangle (black) with incircle (blue), incenter (I), excircles (orange), excenters (JA,JB,JC), internal angle bisectors (red) and external angle bisectors (green). The green triangle is the excentral triangle.
In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter.[1]
An excircle or escribed circle[2] of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two. Every triangle has three distinct excircles, each tangent to one of the triangle's sides.[3]
The center of the incircle, called the incenter, can be found as the intersection of the three internal angle bisectors.[3][4] The center of an excircle is the intersection of the internal bisector of one angle (at vertex A, for example) and the external bisectors of the other two. The center of this excircle is called the excenter relative to the vertex A, or the excenter of A.[3] Because the internal bisector of an angle is perpendicular to its external bisector, it follows that the center of the incircle together with the three excircle centers form an orthocentric system.[5]:p. 182
Polygons with more than three sides do not all have an incircle tangent to all sides; those that do are called tangential polygons. See also Tangent lines to circles.
Contents
[hide]
1 Incircle and incenter
1.1 Incenter
1.1.1 Trilinear coordinates of the incenter
1.1.2 Barycentric coordinates of the incenter
1.1.3 Cartesian coordinates of the incenter
1.1.4 Distances from the incenter to the vertices
1.1.5 Other incenter properties
1.2 Incircle and its radius properties
1.2.1 Distances between vertex and nearest touchpoints
1.2.2 Other incircle properties
1.2.3 Relation to area of the triangle
1.3 Gergonne triangle and point
2 Excircles and excenters
2.1 Trilinear coordinates of excenters
2.2 Exradii
2.2.1 Derivation of exradii formula
2.2.2 Other exradii properties
2.3 Other excircle properties
2.4 Nagel triangle and Nagel point
3 Relation to area of the triangle
3.1 Incircle
3.2 Excircles
4 Related constructions
4.1 Nine-point circle and Feuerbach point
4.2 Incentral and excentral triangles
5 Equations for four circles
6 Euler's theorem
7 Generalization to other polygons
8 See also
9 Notes
10 References
11 External links
11.1 Interactive
Incircle and incenter[edit]
See also: incenter
A triangle, ΔABC, with incircle (blue), incenter (blue, I), contact triangle (red, ΔTaTbTc) and Gergonne point (green, Ge)
Suppose △ A B C {\displaystyle \triangle ABC} has an incircle with radius r and center I. Let a be the length of BC, b the length of AC, and c the length of AB And let T a , T b , and T c {\displaystyle T_{a},T_{b},{\text{ and }}T_{c}} be the touchpoints where the incircle touches BC , AC and AB

Incenter[edit]
The incenter is the point where the internal angle bisectors of ∠ A B C , ∠ B C A , and ∠ B A C {\displaystyle \angle ABC,\angle BCA,{\text{ and }}\angle BAC} meet.

The distance from vertex A to the incenter I is:
d ( A , I ) = c sin  ( 1 2 ∠ B ) cos  ( 1 2 ∠ C ) = b sin  ( 1 2 ∠ C ) cos  ( 1 2 ∠ B ) {\displaystyle d(A,I)=c{\frac {\sin({\frac {1}{2}}\angle B)}{\cos({\frac {1}{2}}\angle C)}}=b{\frac {\sin({\frac {1}{2}}\angle C)}{\cos({\frac {1}{2}}\angle B)}}}

Trilinear coordinates of the incenter[edit]
The trilinear coordinates for a point in the triangle is the ratio of distances to the triangle sides. Because the Incenter is the same distance of all sides the trilinear coordinates for the incenter are [6]
1 : 1 : 1. {\displaystyle \ 1:1:1.}
Barycentric coordinates of the incenter[edit]
The barycentric coordinates for a point in a triangle give weights such that the point is the weighted average of the triangle vertex positions. Barycentric coordinates for the incenter are given by
a : b : c {\displaystyle \ a:b:c}
where a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} are the lengths of the sides of the triangle, or equivalently (using the law of sines) by
sin  ( A ) : sin  ( B ) : sin  ( C ) {\displaystyle \sin(A):\sin(B):\sin(C)}
where A {\displaystyle A} , B {\displaystyle B} , and C {\displaystyle C} are the angles at the three vertices.
Cartesian coordinates of the incenter[edit]
The Cartesian coordinates of the incenter are a weighted average of the coordinates of the three vertices using the side lengths of the triangle relative to the perimeter—i.e., using the barycentric coordinates given above, normalized to sum to unity—as weights. (The weights are positive so the incenter lies inside the triangle as stated above.) If the three vertices are located at ( x a , y a ) {\displaystyle (x_{a},y_{a})} , ( x b , y b ) {\displaystyle (x_{b},y_{b})} , and ( x c , y c ) {\displaystyle (x_{c},y_{c})} , and the sides opposite these vertices have corresponding lengths a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} , then the incenter is at
( a x a + b x b + c x c a + b + c , a y a + b y b + c y c a + b + c ) = a ( x a , y a ) + b ( x b , y b ) + c ( x c , y c ) a + b + c . {\displaystyle {\bigg (}{\frac {ax_{a}+bx_{b}+cx_{c}}{a+b+c}},{\frac {ay_{a}+by_{b}+cy_{c}}{a+b+c}}{\bigg )}={\frac {a(x_{a},y_{a})+b(x_{b},y_{b})+c(x_{c},y_{c})}{a+b+c}}.}
Distances from the incenter to the vertices[edit]
Denoting the incenter of triangle ABC as I, the distances from the incenter to the vertices combined with the lengths of the triangle sides obey the equation[7]
I A ⋅ I A C A ⋅ A B + I B ⋅ I B A B ⋅ B C + I C ⋅ I C B C ⋅ C A = 1. {\displaystyle {\frac {IA\cdot IA}{CA\cdot AB}}+{\frac {IB\cdot IB}{AB\cdot BC}}+{\frac {IC\cdot IC}{BC\cdot CA}}=1.}
Additionally,[8]
I A ⋅ I B ⋅ I C = 4 R r 2 , {\displaystyle IA\cdot IB\cdot IC=4Rr^{2},}
where R and r are the triangle's circumradius and inradius respectively.
Other incenter properties[edit]
The collection of triangle centers may be given the structure of a group under coordinate-wise multiplication of trilinear coordinates; in this group, the incenter forms the identity element.[6]
Incircle and its radius properties[edit]
Distances between vertex and nearest touchpoints[edit]
The distances from a vertex to the two nearest touchpoints are equal:
d ( A , T B ) = d ( A , T C ) ) = 1 2 ( b + c − a ) {\displaystyle d(A,T_{B})=d(A,T_{C}))={\frac {1}{2}}(b+c-a)}
Other incircle properties[edit]
Suppose the tangency points of the incircle divide the sides into lengths of x and y, y and z, and z and x. Then the incircle has the radius[9]
r = x y z x + y + z {\displaystyle r={\sqrt {\frac {xyz}{x+y+z}}}}
and the area of the triangle is
Δ = x y z ( x + y + z ) . {\displaystyle \Delta ={\sqrt {xyz(x+y+z)}}.}
If the altitudes from sides of lengths a, b, and c are ha, hb, and hc then the inradius r is one-third of the harmonic mean of these altitudes, i.e.
r = 1 h a − 1 + h b − 1 + h c − 1 . {\displaystyle r={\frac {1}{h_{a}^{-1}+h_{b}^{-1}+h_{c}^{-1}}}.}
The product of the incircle radius r and the circumcircle radius R of a triangle with sides a, b, and c is[5]:p. 189, #298(d)
r R = a b c 2 ( a + b + c ) . {\displaystyle rR={\frac {abc}{2(a+b+c)}}.}
Some relations among the sides, incircle radius, and circumcircle radius are:[10]
a b + b c + c a = s 2 + ( 4 R + r ) r , {\displaystyle ab+bc+ca=s^{2}+(4R+r)r,}
a 2 + b 2 + c 2 = 2 s 2 − 2 ( 4 R + r ) r . {\displaystyle a^{2}+b^{2}+c^{2}=2s^{2}-2(4R+r)r.}
Any line through a triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter (the center of its incircle). There are either one, two, or three of these for any given triangle.[11]
Denoting the center of the incircle of triangle ABC as I, we have[12]
I A ⋅ I A C A ⋅ A B + I B ⋅ I B A B ⋅ B C + I C ⋅ I C B C ⋅ C A = 1 {\displaystyle {\frac {IA\cdot IA}{CA\cdot AB}}+{\frac {IB\cdot IB}{AB\cdot BC}}+{\frac {IC\cdot IC}{BC\cdot CA}}=1}
and[13]:p.121,#84
I A ⋅ I B ⋅ I C = 4 R r 2 . {\displaystyle IA\cdot IB\cdot IC=4Rr^{2}.}
The distance from any vertex to the incircle tangency on either adjacent side is half the sum of the vertex's adjacent sides minus half the opposite side.[14] Thus for example for vertex B and adjacent tangencies TA and TC,
B T A = B T C = B C + A B − A C 2 . {\displaystyle BT_{A}=BT_{C}={\frac {BC+AB-AC}{2}}.}
The incircle radius is no greater than one-ninth the sum of the altitudes.[15]:p. 289
The squared distance from the incenter I to the circumcenter O is given by[16]:p.232
O I 2 = R ( R − 2 r ) , {\displaystyle OI^{2}=R(R-2r),}
and the distance from the incenter to the center N of the nine point circle is[16]:p.232
I N = 1 2 ( R − 2 r ) < 1 2 R . {\displaystyle IN={\frac {1}{2}}(R-2r)<{\frac {1}{2}}R.}
The incenter lies in the medial triangle (whose vertices are the midpoints of the sides).[16]:p.233, Lemma 1

Relation to area of the triangle[edit]
A triangle, ΔABC, with incircle (blue), incenter (blue, I), contact triangle (red, ΔTaTbTc) and Gergonne point (green, Ge)
The radii of the incircle is related to the area of the triangle.[17] The ratio of the area of the incircle to the area of the triangle is less than or equal to π 3 3 {\displaystyle {\frac {\pi }{3{\sqrt {3}}}}} , with equality holding only for equilateral triangles.[18]

Suppose △ A B C {\displaystyle \triangle ABC} has an incircle with radius r and center I. Let a be the length of BC, b the length of AC, and c the length of AB. Now, the incircle is tangent to AB at some point C′, and so ∠ A T c I {\displaystyle \angle AT_{c}I} is right. Thus the radius TcI is an altitude of △ I A B {\displaystyle \triangle IAB} . Therefore, △ I A B {\displaystyle \triangle IAB} has base length c and height r, and so has area 1 2 c r {\displaystyle {\tfrac {1}{2}}cr} . Similarly, △ I A C {\displaystyle \triangle IAC} has area 1 2 b r {\displaystyle {\tfrac {1}{2}}br} and △ I B C {\displaystyle \triangle IBC} has area 1 2 a r {\displaystyle {\tfrac {1}{2}}ar} . Since these three triangles decompose △ A B C {\displaystyle \triangle ABC} , we see that the area Δ of △ A B C {\displaystyle \Delta {\text{ of }}\triangle ABC} is:
Δ = 1 2 ( a + b + c ) r = s r , {\displaystyle \Delta ={\frac {1}{2}}(a+b+c)r=sr,} and r = Δ s , {\displaystyle r={\frac {\Delta }{s}},}
where Δ {\displaystyle \Delta } is the area of △ A B C {\displaystyle \triangle ABC} and s = 1 2 ( a + b + c ) {\displaystyle s={\frac {1}{2}}(a+b+c)} is its semiperimeter.
For an alternative formula, consider △ I T c A {\displaystyle \triangle IT_{c}A} . This is a right-angled triangle with one side equal to r and the other side equal to r cot  ∠ A 2 {\displaystyle r\cot {\frac {\angle A}{2}}} . The same is true for △ I B ′ A {\displaystyle \triangle IB'A} . The large triangle is composed of 6 such triangles and the total area is:
Δ = r 2 ⋅ ( cot  ∠ A 2 + cot  ∠ B 2 + cot  ∠ C 2 ) {\displaystyle \Delta =r^{2}\cdot (\cot {\frac {\angle A}{2}}+\cot {\frac {\angle B}{2}}+\cot {\frac {\angle C}{2}})}
Gergonne triangle and point[edit]
A triangle, ΔABC, with incircle (blue), incenter (blue, I), contact triangle (red, ΔTaTbTc) and Gergonne point (green, Ge)
The Gergonne triangle (of ABC) is defined by the 3 touchpoints of the incircle on the 3 sides. The touchpoint opposite A is denoted TA, etc.
This Gergonne triangle TATBTC is also known as the contact triangle or intouch triangle of ABC. Its area is
K T = K 2 r 2 s a b c {\displaystyle K_{T}=K{\frac {2r^{2}s}{abc}}}
where K {\displaystyle K} , r {\displaystyle r} , s {\displaystyle s} are the area, radius of the incircle and semiperimeter of the original triangle, and a {\displaystyle a} , b {\displaystyle b} , c {\displaystyle c} are the side lengths of the original triangle. This is the same area as that of the extouch triangle.[19]
The three lines ATA, BTB and CTC intersect in a single point called the Gergonne point, denoted as Ge - X(7). The Gergonne point lies in the open orthocentroidal disk punctured at its own center, and could be any point therein.[20]
Interestingly, the Gergonne point of a triangle is the symmedian point of the Gergonne triangle. For a full set of properties of the Gergonne point see.[21]
Trilinear coordinates for the vertices of the intouch triangle are given by
vertex T A = 0 : sec 2  ( B 2 ) : sec 2  ( C 2 ) {\displaystyle {\text{vertex}}\,T_{A}=0:\sec ^{2}\left({\frac {B}{2}}\right):\sec ^{2}\left({\frac {C}{2}}\right)}
vertex T B = sec 2  ( A 2 ) : 0 : sec 2  ( C 2 ) {\displaystyle {\text{vertex}}\,T_{B}=\sec ^{2}\left({\frac {A}{2}}\right):0:\sec ^{2}\left({\frac {C}{2}}\right)}
vertex T C = sec 2  ( A 2 ) : sec 2  ( B 2 ) : 0 {\displaystyle {\text{vertex}}\,T_{C}=\sec ^{2}\left({\frac {A}{2}}\right):\sec ^{2}\left({\frac {B}{2}}\right):0}
Trilinear coordinates for the Gergonne point are given by
sec 2  ( A 2 ) : sec 2  ( B 2 ) : sec 2  ( C 2 ) {\displaystyle \sec ^{2}\left({\frac {A}{2}}\right):\sec ^{2}\left({\frac {B}{2}}\right):\sec ^{2}\left({\frac {C}{2}}\right)} ,
or, equivalently, by the Law of Sines,
b c b + c − a : c a c + a − b : a b a + b − c {\displaystyle {\frac {bc}{b+c-a}}:{\frac {ca}{c+a-b}}:{\frac {ab}{a+b-c}}} .
Excircles and excenters[edit]
A triangle (black) with incircle (blue), incenter (I), excircles (orange), excenters (JA,JB,JC), internal angle bisectors (red) and external angle bisectors (green). The green triangle is the excentral triangle.
An excircle or escribed circle[22] of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two. Every triangle has three distinct excircles, each tangent to one of the triangle's sides.[3]
The center of an excircle is the intersection of the internal bisector of one angle (at vertex A, for example) and the external bisectors of the other two. The center of this excircle is called the excenter relative to the vertex A, or the excenter of A.[3] Because the internal bisector of an angle is perpendicular to its external bisector, it follows that the center of the incircle together with the three excircle centers form an orthocentric system.[5]:p. 182
Trilinear coordinates of excenters[edit]
While A triangle's incenter has trilinear coordinates 1 : 1 : 1. {\displaystyle 1:1:1.} Its excenters (the centers of its excircles) have trilinears − 1 : 1 : 1 {\displaystyle -1:1:1} , 1 : − 1 : 1 {\displaystyle 1:-1:1} and 1 : 1 : − 1. {\displaystyle 1:1:-1.}
Exradii[edit]
The radii of the excircles are called the exradii.
the exradius of the Excircle opposite A (so touching BC, centered at J A {\displaystyle J_{A}} is
r a = s ( s − b ) ( s − c ) s − a . {\displaystyle r_{a}={\sqrt {\frac {s(s-b)(s-c)}{s-a}}}.} [23]
where s = 1 2 ( a + b + c ) {\displaystyle s={\frac {1}{2}}(a+b+c)}
Derivation of exradii formula[edit]
Let the excircle at side AB touch at side AC extended at G, and let this excircle's radius be r c {\displaystyle r_{c}} and its center be J c {\displaystyle J_{c}} .
Then J c G {\displaystyle J_{c}G} is an altitude of △ A C J c {\displaystyle \triangle ACJ_{c}} , so △ A C J c {\displaystyle \triangle ACJ_{c}} has area 1 2 b r c {\displaystyle {\tfrac {1}{2}}br_{c}} . By a similar argument, △ B C J c {\displaystyle \triangle BCJ_{c}} has area 1 2 a r c {\displaystyle {\tfrac {1}{2}}ar_{c}} and △ A B J c {\displaystyle \triangle ABJ_{c}} has area 1 2 c r c {\displaystyle {\tfrac {1}{2}}cr_{c}} . Thus the area Δ {\displaystyle \Delta } of triangle △ A B C {\displaystyle \triangle ABC} is
Δ = 1 2 ( a + b − c ) r c = ( s − c ) r c {\displaystyle \Delta ={\frac {1}{2}}(a+b-c)r_{c}=(s-c)r_{c}} .
So, by symmetry, denoting r {\displaystyle r} as the radius of the incircle,
Δ = s r = ( s − a ) r a = ( s − b ) r b = ( s − c ) r c {\displaystyle \Delta =sr=(s-a)r_{a}=(s-b)r_{b}=(s-c)r_{c}} .
By the Law of Cosines, we have
cos  A = b 2 + c 2 − a 2 2 b c {\displaystyle \cos A={\frac {b^{2}+c^{2}-a^{2}}{2bc}}}
Combining this with the identity sin 2  A + cos 2  A = 1 {\displaystyle \sin ^{2}A+\cos ^{2}A=1} , we have
sin  A = − a 4 − b 4 − c 4 + 2 a 2 b 2 + 2 b 2 c 2 + 2 a 2 c 2 2 b c {\displaystyle \sin A={\frac {\sqrt {-a^{4}-b^{4}-c^{4}+2a^{2}b^{2}+2b^{2}c^{2}+2a^{2}c^{2}}}{2bc}}}
But Δ = 1 2 b c sin  A {\displaystyle \Delta ={\tfrac {1}{2}}bc\sin A} , and so
Δ = 1 4 − a 4 − b 4 − c 4 + 2 a 2 b 2 + 2 b 2 c 2 + 2 a 2 c 2 = 1 4 ( a + b + c ) ( − a + b + c ) ( a − b + c ) ( a + b − c ) = s ( s − a ) ( s − b ) ( s − c ) , {\displaystyle {\begin{aligned}\Delta &={\frac {1}{4}}{\sqrt {-a^{4}-b^{4}-c^{4}+2a^{2}b^{2}+2b^{2}c^{2}+2a^{2}c^{2}}}\\&={\frac {1}{4}}{\sqrt {(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}}\\&={\sqrt {s(s-a)(s-b)(s-c)}},\end{aligned}}}
which is Heron's formula.
Combining this with s r = Δ {\displaystyle sr=\Delta } , we have
r 2 = Δ 2 s 2 = ( s − a ) ( s − b ) ( s − c ) s . {\displaystyle r^{2}={\frac {\Delta ^{2}}{s^{2}}}={\frac {(s-a)(s-b)(s-c)}{s}}.}
Similarly, ( s − a ) r a = Δ {\displaystyle (s-a)r_{a}=\Delta } gives
r a 2 = s ( s − b ) ( s − c ) s − a {\displaystyle r_{a}^{2}={\frac {s(s-b)(s-c)}{s-a}}}
and
r a = s ( s − b ) ( s − c ) s − a . {\displaystyle r_{a}={\sqrt {\frac {s(s-b)(s-c)}{s-a}}}.} [24]
Other exradii properties[edit]
From the formulas above one can see that the excircles are always larger than the incircle and that the largest excircle is the one tangent to the longest side and the smallest excircle is tangent to the shortest side. Further, combining these formulas yields:[25]
Δ = r r a r b r c . {\displaystyle \Delta ={\sqrt {rr_{a}r_{b}r_{c}}}.}
The ratio of the area of the incircle to the area of the triangle is less than or equal to π 3 3 {\displaystyle {\frac {\pi }{3{\sqrt {3}}}}} , with equality holding only for equilateral triangles.[26]
Other excircle properties[edit]
The circular hull of the excircles is internally tangent to each of the excircles, and thus is an Apollonius circle.[27] The radius of this Apollonius circle is r 2 + s 2 4 r {\displaystyle {\frac {r^{2}+s^{2}}{4r}}} where r is the incircle radius and s is the semiperimeter of the triangle.[28]
The following relations hold among the inradius r, the circumradius R, the semiperimeter s, and the excircle radii ra, rb, rc:[10]
r a + r b + r c = 4 R + r , {\displaystyle r_{a}+r_{b}+r_{c}=4R+r,}
r a r b + r b r c + r c r a = s 2 , {\displaystyle r_{a}r_{b}+r_{b}r_{c}+r_{c}r_{a}=s^{2},}
r a 2 + r b 2 + r c 2 = ( 4 R + r ) 2 − 2 s 2 , {\displaystyle r_{a}^{2}+r_{b}^{2}+r_{c}^{2}=(4R+r)^{2}-2s^{2},}
The circle through the centers of the three excircles has radius 2R.[10]
If H is the orthocenter of triangle ABC, then[10]
r a + r b + r c + r = A H + B H + C H + 2 R , {\displaystyle r_{a}+r_{b}+r_{c}+r=AH+BH+CH+2R,}
r a 2 + r b 2 + r c 2 + r 2 = A H 2 + B H 2 + C H 2 + ( 2 R ) 2 . {\displaystyle r_{a}^{2}+r_{b}^{2}+r_{c}^{2}+r^{2}=AH^{2}+BH^{2}+CH^{2}+(2R)^{2}.}
Nagel triangle and Nagel point[edit]
Main article: Extouch triangle
The extouch triangle (ΔTATBTC, with red boundary) and the Nagel point (blue, N) of a triangle (ΔABC, with black boundary). The orange circles are the excircles of the triangle.
The Nagel triangle or extouch triangle of ABC is denoted by the vertices TA, TB and TC that are the three points where the excircles touch the reference triangle ABC and where TA is opposite of A, etc. This triangle TATBTC is also known as the extouch triangle of ABC. The circumcircle of the extouch triangle TATBTC is called the Mandart circle.
The three lines ATA, BTB and CTC are called the splitters of the triangle; they each bisect the perimeter of the triangle,
A B + B T A = A C + C T A = 1 2 ( A B + B C + A C ) {\displaystyle AB+BT_{A}=AC+CT_{A}={\frac {1}{2}}\left(AB+BC+AC\right)}
The splitters intersect in a single point, the triangle's Nagel point Na - X(8).
Trilinear coordinates for the vertices of the extouch triangle are given by
vertex T A = 0 : csc 2  ( B 2 ) : csc 2  ( C 2 ) {\displaystyle {\text{vertex}}\,T_{A}=0:\csc ^{2}\left({\frac {B}{2}}\right):\csc ^{2}\left({\frac {C}{2}}\right)}
vertex T B = csc 2  ( A 2 ) : 0 : csc 2  ( C 2 ) {\displaystyle {\text{vertex}}\,T_{B}=\csc ^{2}\left({\frac {A}{2}}\right):0:\csc ^{2}\left({\frac {C}{2}}\right)}
vertex T C = csc 2  ( A 2 ) : csc 2  ( B 2 ) : 0 {\displaystyle {\text{vertex}}\,T_{C}=\csc ^{2}\left({\frac {A}{2}}\right):\csc ^{2}\left({\frac {B}{2}}\right):0}
Trilinear coordinates for the Nagel point are given by
csc 2  ( A 2 ) : csc 2  ( B 2 ) : csc 2  ( C 2 ) {\displaystyle \csc ^{2}\left({\frac {A}{2}}\right):\csc ^{2}\left({\frac {B}{2}}\right):\csc ^{2}\left({\frac {C}{2}}\right)} ,
or, equivalently, by the Law of Sines,
b + c − a a : c + a − b b : a + b − c c {\displaystyle {\frac {b+c-a}{a}}:{\frac {c+a-b}{b}}:{\frac {a+b-c}{c}}} .
It is the isotomic conjugate of the Gergonne point.
Relation to area of the triangle[edit]
The radii of the incircles and excircles are closely related to the area of the triangle.[29]
Incircle[edit]
Suppose △ A B C {\displaystyle \triangle ABC} has an incircle with radius r and center I. Let a be the length of BC, b the length of AC, and c the length of AB. Now, the incircle is tangent to AB at some point C′, and so ∠ A C ′ I {\displaystyle \angle AC'I} is right. Thus the radius C'I is an altitude of △ I A B {\displaystyle \triangle IAB} . Therefore, △ I A B {\displaystyle \triangle IAB} has base length c and height r, and so has area 1 2 c r {\displaystyle {\tfrac {1}{2}}cr} . Similarly, △ I A C {\displaystyle \triangle IAC} has area 1 2 b r {\displaystyle {\tfrac {1}{2}}br} and △ I B C {\displaystyle \triangle IBC} has area 1 2 a r {\displaystyle {\tfrac {1}{2}}ar} . Since these three triangles decompose △ A B C {\displaystyle \triangle ABC} , we see that
Δ = 1 2 ( a + b + c ) r = s r , {\displaystyle \Delta ={\frac {1}{2}}(a+b+c)r=sr,} and r = Δ s , {\displaystyle r={\frac {\Delta }{s}},}
where Δ {\displaystyle \Delta } is the area of △ A B C {\displaystyle \triangle ABC} and s = 1 2 ( a + b + c ) {\displaystyle s={\frac {1}{2}}(a+b+c)} is its semiperimeter.
For an alternative formula, consider △ I C ′ A {\displaystyle \triangle IC'A} . This is a right-angled triangle with one side equal to r and the other side equal to r cot  ∠ A 2 {\displaystyle r\cot {\frac {\angle A}{2}}} . The same is true for △ I B ′ A {\displaystyle \triangle IB'A} . The large triangle is composed of 6 such triangles and the total area is:
Δ = r 2 ⋅ ( cot  ∠ A 2 + cot  ∠ B 2 + cot  ∠ C 2 ) {\displaystyle \Delta =r^{2}\cdot (\cot {\frac {\angle A}{2}}+\cot {\frac {\angle B}{2}}+\cot {\frac {\angle C}{2}})}
Excircles[edit]
The radii of the excircles are called the exradii. Let the excircle at side AB touch at side AC extended at G, and let this excircle's radius be r c {\displaystyle r_{c}} and its center be I c {\displaystyle I_{c}} . Then I c G {\displaystyle I_{c}G} is an altitude of △ A C I c {\displaystyle \triangle ACI_{c}} , so △ A C I c {\displaystyle \triangle ACI_{c}} has area 1 2 b r c {\displaystyle {\tfrac {1}{2}}br_{c}} . By a similar argument, △ B C I c {\displaystyle \triangle BCI_{c}} has area 1 2 a r c {\displaystyle {\tfrac {1}{2}}ar_{c}} and △ A B I c {\displaystyle \triangle ABI_{c}} has area 1 2 c r c {\displaystyle {\tfrac {1}{2}}cr_{c}} . Thus
Δ = 1 2 ( a + b − c ) r c = ( s − c ) r c {\displaystyle \Delta ={\frac {1}{2}}(a+b-c)r_{c}=(s-c)r_{c}} .
So, by symmetry,
Δ = s r = ( s − a ) r a = ( s − b ) r b = ( s − c ) r c {\displaystyle \Delta =sr=(s-a)r_{a}=(s-b)r_{b}=(s-c)r_{c}} .
By the Law of Cosines, we have
cos  A = b 2 + c 2 − a 2 2 b c {\displaystyle \cos A={\frac {b^{2}+c^{2}-a^{2}}{2bc}}}
Combining this with the identity sin 2  A + cos 2  A = 1 {\displaystyle \sin ^{2}A+\cos ^{2}A=1} , we have
sin  A = − a 4 − b 4 − c 4 + 2 a 2 b 2 + 2 b 2 c 2 + 2 a 2 c 2 2 b c {\displaystyle \sin A={\frac {\sqrt {-a^{4}-b^{4}-c^{4}+2a^{2}b^{2}+2b^{2}c^{2}+2a^{2}c^{2}}}{2bc}}}
But Δ = 1 2 b c sin  A {\displaystyle \Delta ={\tfrac {1}{2}}bc\sin A} , and so
Δ = 1 4 − a 4 − b 4 − c 4 + 2 a 2 b 2 + 2 b 2 c 2 + 2 a 2 c 2 = 1 4 ( a + b + c ) ( − a + b + c ) ( a − b + c ) ( a + b − c ) = s ( s − a ) ( s − b ) ( s − c ) , {\displaystyle {\begin{aligned}\Delta &={\frac {1}{4}}{\sqrt {-a^{4}-b^{4}-c^{4}+2a^{2}b^{2}+2b^{2}c^{2}+2a^{2}c^{2}}}\\&={\frac {1}{4}}{\sqrt {(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}}\\&={\sqrt {s(s-a)(s-b)(s-c)}},\end{aligned}}}
which is Heron's formula.
Combining this with s r = Δ {\displaystyle sr=\Delta } , we have
r 2 = Δ 2 s 2 = ( s − a ) ( s − b ) ( s − c ) s . {\displaystyle r^{2}={\frac {\Delta ^{2}}{s^{2}}}={\frac {(s-a)(s-b)(s-c)}{s}}.}
Similarly, ( s − a ) r a = Δ {\displaystyle (s-a)r_{a}=\Delta } gives
r a 2 = s ( s − b ) ( s − c ) s − a {\displaystyle r_{a}^{2}={\frac {s(s-b)(s-c)}{s-a}}}
and
r a = s ( s − b ) ( s − c ) s − a . {\displaystyle r_{a}={\sqrt {\frac {s(s-b)(s-c)}{s-a}}}.} [30]
From these formulas one can see that the excircles are always larger than the incircle and that the largest excircle is the one tangent to the longest side and the smallest excircle is tangent to the shortest side. Further, combining these formulas yields:[31]
Δ = r r a r b r c . {\displaystyle \Delta ={\sqrt {rr_{a}r_{b}r_{c}}}.}
The ratio of the area of the incircle to the area of the triangle is less than or equal to π 3 3 {\displaystyle {\frac {\pi }{3{\sqrt {3}}}}} , with equality holding only for equilateral triangles.[32]
Related constructions[edit]
Nine-point circle and Feuerbach point[edit]
Main article: Nine-point circle
The nine-point circle is tangent to the incircle and excircles
In geometry, the nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant concyclic points defined from the triangle. These nine points are:
The midpoint of each side of the triangle
The foot of each altitude
The midpoint of the line segment from each vertex of the triangle to the orthocenter (where the three altitudes meet; these line segments lie on their respective altitudes).[33][34]
In 1822 Karl Feuerbach discovered that any triangle's nine-point circle is externally tangent to that triangle's three excircles and internally tangent to its incircle; this result is known as Feuerbach's theorem. He proved that:
... the circle which passes through the feet of the altitudes of a triangle is tangent to all four circles which in turn are tangent to the three sides of the triangle... (Feuerbach 1822)
The triangle center at which the incircle and the nine-point circle touch is called the Feuerbach point.
Incentral and excentral triangles[edit]
The points of intersection of the interior angle bisectors of ABC with the segments BC, CA, AB are the vertices of the incentral triangle. Trilinear coordinates for the vertices of the incentral triangle are given by
( vertex opposite A ) = 0 : 1 : 1 {\displaystyle \ ({\text{vertex opposite}}\,A)=0:1:1}
( vertex opposite B ) = 1 : 0 : 1 {\displaystyle \ ({\text{vertex opposite}}\,B)=1:0:1}
( vertex opposite C ) = 1 : 1 : 0 {\displaystyle \ ({\text{vertex opposite}}\,C)=1:1:0}
The excentral triangle of a reference triangle has vertices at the centers of the reference triangle's excircles. Its sides are on the external angle bisectors of the reference triangle (see figure at top of page). Trilinear coordinates for the vertices of the excentral triangle are given by
( vertex opposite A ) = − 1 : 1 : 1 {\displaystyle ({\text{vertex opposite}}\,A)=-1:1:1}
( vertex opposite B ) = 1 : − 1 : 1 {\displaystyle ({\text{vertex opposite}}\,B)=1:-1:1}
( vertex opposite C ) = 1 : 1 : − 1 {\displaystyle ({\text{vertex opposite}}\,C)=1:1:-1}
Equations for four circles[edit]
Let x : y : z be a variable point in trilinear coordinates, and let u = cos2(A/2), v = cos2(B/2), w = cos2(C/2). The four circles described above are given equivalently by either of the two given equations:[35]:p. 210–215
Incircle:
u 2 x 2 + v 2 y 2 + w 2 z 2 − 2 v w y z − 2 w u z x − 2 u v x y = 0 {\displaystyle \ u^{2}x^{2}+v^{2}y^{2}+w^{2}z^{2}-2vwyz-2wuzx-2uvxy=0}
± x cos  A 2 ± y cos  B 2 ± z cos  C 2 = 0 {\displaystyle \pm {\sqrt {x}}\cos {\frac {A}{2}}\pm {\sqrt {y}}\cos {\frac {B}{2}}\pm {\sqrt {z}}\cos {\frac {C}{2}}=0}
A-excircle:
u 2 x 2 + v 2 y 2 + w 2 z 2 − 2 v w y z + 2 w u z x + 2 u v x y = 0 {\displaystyle \ u^{2}x^{2}+v^{2}y^{2}+w^{2}z^{2}-2vwyz+2wuzx+2uvxy=0}
± − x cos  A 2 ± y cos  B 2 ± z cos  C 2 = 0 {\displaystyle \pm {\sqrt {-x}}\cos {\frac {A}{2}}\pm {\sqrt {y}}\cos {\frac {B}{2}}\pm {\sqrt {z}}\cos {\frac {C}{2}}=0}
B-excircle:
u 2 x 2 + v 2 y 2 + w 2 z 2 + 2 v w y z − 2 w u z x + 2 u v x y = 0 {\displaystyle \ u^{2}x^{2}+v^{2}y^{2}+w^{2}z^{2}+2vwyz-2wuzx+2uvxy=0}
± x cos  A 2 ± − y cos  B 2 ± z cos  C 2 = 0 {\displaystyle \pm {\sqrt {x}}\cos {\frac {A}{2}}\pm {\sqrt {-y}}\cos {\frac {B}{2}}\pm {\sqrt {z}}\cos {\frac {C}{2}}=0}
C-excircle:
u 2 x 2 + v 2 y 2 + w 2 z 2 + 2 v w y z + 2 w u z x − 2 u v x y = 0 {\displaystyle \ u^{2}x^{2}+v^{2}y^{2}+w^{2}z^{2}+2vwyz+2wuzx-2uvxy=0}
± x cos  A 2 ± y cos  B 2 ± − z cos  C 2 = 0 {\displaystyle \pm {\sqrt {x}}\cos {\frac {A}{2}}\pm {\sqrt {y}}\cos {\frac {B}{2}}\pm {\sqrt {-z}}\cos {\frac {C}{2}}=0}
Euler's theorem[edit]
Euler's theorem states that in a triangle:
( R − r ) 2 = d 2 + r 2 , {\displaystyle (R-r)^{2}=d^{2}+r^{2},}
where R and r are the circumradius and inradius respectively, and d is the distance between the circumcenter and the incenter.
For excircles the equation is similar:
( R + r e x ) 2 = d e x 2 + r e x 2 , {\displaystyle (R+r_{ex})^{2}=d_{ex}^{2}+r_{ex}^{2},}
where rex is the radius of one of the excircles, and dex is the distance between the circumcenter and this excircle's center.[36][37][38]
Generalization to other polygons[edit]
Some (but not all) quadrilaterals have an incircle. These are called tangential quadrilaterals. Among their many properties perhaps the most important is that their two pairs of opposite sides have equal sums. This is called the Pitot theorem.
More generally, a polygon with any number of sides that has an inscribed circle—one that is tangent to each side—is called a tangential polygon.
See also[edit]
Circumgon
Circumscribed circle
Ex-tangential quadrilateral
Harcourt's theorem
Inconic
Inscribed sphere
Power of a point
Steiner inellipse
Tangential quadrilateral
Trillium theorem
Notes[edit]
Jump up ^ Kay (1969, p. 140)
Jump up ^ Altshiller-Court (1925, p. 74)
^ Jump up to: a b c d e Altshiller-Court (1925, p. 73)
Jump up ^ Kay (1969, p. 117)
^ Jump up to: a b c Johnson, Roger A., Advanced Euclidean Geometry, Dover, 2007 (orig. 1929).
^ Jump up to: a b Encyclopedia of Triangle Centers Archived 2012-04-19 at the Wayback Machine., accessed 2014-10-28.
Jump up ^ Allaire, Patricia R.; Zhou, Junmin; Yao, Haishen (March 2012), "Proving a nineteenth century ellipse identity", Mathematical Gazette, 96: 161–165 .
Jump up ^ Altshiller-Court, Nathan (1980), College Geometry, Dover Publications . #84, p. 121.
Jump up ^ Chu, Thomas, The Pentagon, Spring 2005, p. 45, problem 584.
^ Jump up to: a b c d Bell, Amy, "Hansen’s right triangle theorem, its converse and a generalization", Forum Geometricorum 6, 2006, 335–342.
Jump up ^ Kodokostas, Dimitrios, "Triangle Equalizers," Mathematics Magazine 83, April 2010, pp. 141-146.
Jump up ^ Allaire, Patricia R.; Zhou, Junmin; and Yao, Haishen, "Proving a nineteenth century ellipse identity", Mathematical Gazette 96, March 2012, 161-165.
Jump up ^ Altshiller-Court, Nathan. College Geometry, Dover Publications, 1980.
Jump up ^ Mathematical Gazette, July 2003, 323-324.
Jump up ^ Posamentier, Alfred S., and Lehmann, Ingmar. The Secrets of Triangles, Prometheus Books, 2012.
^ Jump up to: a b c Franzsen, William N. (2011). "The distance from the incenter to the Euler line" (PDF). Forum Geometricorum. 11: 231–236. MR 2877263. .
Jump up ^ Coxeter, H.S.M. "Introduction to Geometry 2nd ed. Wiley, 1961.
Jump up ^ Minda, D., and Phelps, S., "Triangles, ellipses, and cubic polynomials", American Mathematical Monthly 115, October 2008, 679-689: Theorem 4.1.
Jump up ^ Weisstein, Eric W. "Contact Triangle." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ContactTriangle.html
Jump up ^ Christopher J. Bradley and Geoff C. Smith, "The locations of triangle centers", Forum Geometricorum 6 (2006), 57--70. http://forumgeom.fau.edu/FG2006volume6/FG200607index.html
Jump up ^ Dekov, Deko (2009). "Computer-generated Mathematics : The Gergonne Point" (PDF). Journal of Computer-generated Euclidean Geometry. 1: 1–14. Archived from the original (PDF) on 2010-11-05.
Jump up ^ Altshiller-Court (1925, p. 74)
Jump up ^ Altshiller-Court (1925, p. 79)
Jump up ^ Altshiller-Court (1925, p. 79)
Jump up ^ Baker, Marcus, "A collection of formulae for the area of a plane triangle," Annals of Mathematics, part 1 in vol. 1(6), January 1885, 134-138. (See also part 2 in vol. 2(1), September 1885, 11-18.)
Jump up ^ Minda, D., and Phelps, S., "Triangles, ellipses, and cubic polynomials", American Mathematical Monthly 115, October 2008, 679-689: Theorem 4.1.
Jump up ^ Grinberg, Darij, and Yiu, Paul, "The Apollonius Circle as a Tucker Circle", Forum Geometricorum 2, 2002: pp. 175-182.
Jump up ^ Stevanovi´c, Milorad R., "The Apollonius circle and related triangle centers", Forum Geometricorum 3, 2003, 187-195.
Jump up ^ Coxeter, H.S.M. "Introduction to Geometry 2nd ed. Wiley, 1961.
Jump up ^ Altshiller-Court (1925, p. 79)
Jump up ^ Baker, Marcus, "A collection of formulae for the area of a plane triangle," Annals of Mathematics, part 1 in vol. 1(6), January 1885, 134-138. (See also part 2 in vol. 2(1), September 1885, 11-18.)
Jump up ^ Minda, D., and Phelps, S., "Triangles, ellipses, and cubic polynomials", American Mathematical Monthly 115, October 2008, 679-689: Theorem 4.1.
Jump up ^ Altshiller-Court (1925, pp. 103–110)
Jump up ^ Kay (1969, pp. 18,245)
Jump up ^ Whitworth, William Allen. Trilinear Coordinates and Other Methods of Modern Analytical Geometry of Two Dimensions, Forgotten Books, 2012 (orig. Deighton, Bell, and Co., 1866). http://www.forgottenbooks.com/search?q=Trilinear+coordinates&t=books
Jump up ^ Nelson, Roger, "Euler's triangle inequality via proof without words," Mathematics Magazine 81(1), February 2008, 58-61.
Jump up ^ Johnson, R. A. Modern Geometry, Houghton Mifflin, Boston, 1929: p. 187.
Jump up ^ Emelyanov, Lev, and Emelyanova, Tatiana. "Euler’s formula and Poncelet’s porism", Forum Geometricorum 1, 2001: pp. 137–140.
References[edit]
Altshiller-Court, Nathan (1925), College Geometry: An Introduction to the Modern Geometry of the Triangle and the Circle (2nd ed.), New York: Barnes & Noble, LCCN 52013504
Kay, David C. (1969), College Geometry, New York: Holt, Rinehart and Winston, LCCN 69012075
Kimberling, Clark (1998). "Triangle Centers and Central Triangles". Congressus Numerantium (129): i–xxv,1–295.
Kiss, Sándor (2006). "The Orthic-of-Intouch and Intouch-of-Orthic Triangles". Forum Geometricorum (6): 171–177.
External links[edit]
Derivation of formula for radius of incircle of a triangle
Weisstein, Eric W. "Incircle". MathWorld.
Interactive[edit]
Triangle incenter Triangle incircle Incircle of a regular polygon With interactive animations
Constructing a triangle's incenter / incircle with compass and straightedge An interactive animated demonstration
Equal Incircles Theorem at cut-the-knot
Five Incircles Theorem at cut-the-knot
Pairs of Incircles in a Quadrilateral at cut-the-knot
An interactive Java applet for the incenter
Retrieved from "https://en.wikipedia.org/w/index.php?title=Incircle_and_excircles_of_a_triangle&oldid=809993839"
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Law of the instrument
From Wikipedia, the free encyclopedia
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The concept known as the law of the instrument, otherwise known as the law of the hammer,[1] Maslow's hammer (or gavel), or the golden hammer,[a] is a cognitive bias that involves an over-reliance on a familiar tool. As Abraham Maslow said in 1966, "I suppose it is tempting, if the only tool you have is a hammer, to treat everything as if it were a nail."[2]
The concept is attributed both to Maslow[3] and to Abraham Kaplan,[4][5] although the hammer and nail line may not be original to either of them. It has in fact been attributed "to everyone from Buddha to Bernard Baruch". Mark Twain has sometimes been credited with it, though it cannot be found in Twain's published writings.[6] Under the name of "Baruch's Observation", it has also been attributed to the stock market speculator and author Bernard M. Baruch.[7]
Sharlyn Lauby has drawn the following lesson from the law: "We need choose the tools we work with carefully." Some tools are adaptable, while others should be employed "only for their intended purpose".[8]
Contents
[hide]
1 History
2 Abraham Kaplan
3 Tomkins and Colby
4 Abraham Maslow
5 Robert Kagan
6 Lee Loevinger
7 Manifestations
7.1 Psychiatry
7.2 Computer programming
7.3 Education
8 Related concepts
9 See also
10 Notes
11 References
History[edit]
The English expression "a Birmingham screwdriver" meaning a hammer, references the habit of using the one tool for all purposes, and predates both Kaplan and Maslow by at least a century.[9]
In 1868, a London periodical, Once a Week, contained this observation: "Give a boy a hammer and chisel; show him how to use them; at once he begins to hack the doorposts, to take off the corners of shutter and window frames, until you teach him a better use for them, and how to keep his activity within bounds."[10]
Abraham Kaplan[edit]
The first recorded statement of the concept was Abraham Kaplan's, in 1964: "I call it the law of the instrument, and it may be formulated as follows: Give a small boy a hammer, and he will find that everything he encounters needs pounding."[11]
In February 1962, Abraham Kaplan, then a Professor of Philosophy, gave a banquet speech at a conference of the American Educational Research Association that was being held at UCLA. An article in the June 1962 issue of the Journal of Medical Education stated that "the highlight of the 3-day meeting ... was to be found in Kaplan's comment on the choice of methods for research. He urged that scientists exercise good judgment in the selection of appropriate methods for their research. Because certain methods happen to be handy, or a given individual has been trained to use a specific method, is no assurance that the method is appropriate for all problems. He cited Kaplan’s Law of the Instrument: 'Give a boy a hammer and everything he meets has to be pounded.'"
In The Conduct of Inquiry: Methodology for Behavioral Science (1964), Kaplan again mentioned the law of the instrument saying, "It comes as no particular surprise to discover that a scientist formulates problems in a way which requires for their solution just those techniques in which he himself is especially skilled." And in a 1964 article for The Library Quarterly, he again cited the law and commented: "We tend to formulate our problems in such a way as to make it seem that the solutions to those problems demand precisely what we already happen to have at hand."[10]
Tomkins and Colby[edit]
In a 1963 essay collection, Computer Simulation of Personality: Frontier of Psychological Theory, Silvan Tomkins wrote about "the tendency of jobs to be adapted to tools, rather than adapting tools to jobs". He wrote: "If one has a hammer one tends to look for nails, and if one has a computer with a storage capacity, but no feelings, one is more likely to concern oneself with remembering and with problem solving than with loving and hating." In the same book, Kenneth Mark Colby explicitly cited the law, writing: "The First Law of the Instrument states that if you give a boy a hammer, he suddenly finds that everything needs pounding. The computer program may be our current hammer, but it must be tried. One cannot decide from purely armchair considerations whether or not it will be of any value."[10]
Abraham Maslow[edit]
Maslow's hammer, popularly phrased as "if all you have is a hammer, everything looks like a nail" and variants thereof, is from Abraham Maslow's The Psychology of Science, published in 1966. Abraham Maslow wrote: "I remember seeing an elaborate and complicated automatic washing machine for automobiles that did a beautiful job of washing them. But it could do only that, and everything else that got into its clutches was treated as if it were an automobile to be washed. I suppose it is tempting, if the only tool you have is a hammer, to treat everything as if it were a nail."[10][2]
Robert Kagan[edit]
In his 2003 book, Of Paradise and Power, historian Robert Kagan suggested a corollary to the law: "When you don't have a hammer, you don't want anything to look like a nail." According to Kagan, the corollary explains the difference in views on the use of military force the United States and Europe have held since the end of World War II.[12]
Lee Loevinger[edit]
In 1967, Lee Loevinger of the Federal Communications Commission dubbed the law "Loevinger's law of irresistible use", and applied it to government: "The political science analogue is that if there is a government agency, this proves something needs regulating."
In 1984, investor Warren Buffett criticized academic studies of financial markets that made use of inappropriate mathematical approaches: "It isn't necessarily because such studies have any utility; it's simply that the data are there and academicians have worked hard to learn the mathematical skills needed to manipulate them. Once these skills are acquired, it seems sinful not to use them, even if the usage has no utility or negative utility. As a friend said, to a man with a hammer, everything looks like a nail."[10]
Manifestations[edit]
Psychiatry[edit]
One application the law of the instrument has been observed in is the prescribing usage of anti-psychotic drugs. During Maslow's era, the only medicines available for psychiatric patients were the anti-psychotics stelazine and thorazine, so it was often the case that other mental illnesses were treated as if they were psychoses.[citation needed]
Computer programming[edit]
The notion of a golden hammer, "a familiar technology or concept applied obsessively to many software problems", was introduced into information technology literature in 1998 as an anti-pattern: a programming practice to be avoided.[13]
Software developer José M. Gilgado has written that the law is still relevant in the 21st century and is highly applicable to software development. Many times software developers, he observed, "tend to use the same known tools to do a completely new different project with new constraints". He blamed this on "the comfort zone state where you don't change anything to avoid risk. The problem with using the same tools every time you can is that you don't have enough arguments to make a choice because you have nothing to compare to and is limiting your knowledge." The solution is "to keep looking for the best possible choice, even if we aren't very familiar with it". This includes using a computer language with which one is unfamiliar. He noted that the product RubyMotion enables developers to "wrap" unknown computer languages in a familiar computer language and thus avoid having to learn them. But Gilgado found this approach inadvisable, because it reinforces the habit of avoiding new tools.[14]
Education[edit]
One observer stated in 2016 that the law of the instrument may be "the least discussed law when talking about education", but is "the most important to warn for in educational discussions". He asked: "How many times do you read: this tool will change everything in education? We know that it won’t, but most of the time the person expressing this is really convinced of his hammer." Some educators say: "let’s teach all children how to code".[15]
Related concepts[edit]
Other forms of narrow-minded instrumentalism[16] include: déformation professionnelle, a French term for "looking at things from the point of view of one's profession", and regulatory capture, the tendency for regulators to look at things from the point of view of the profession they are regulating.
See also[edit]
Confirmation bias
Einstellung effect
Panacea
Notes[edit]
Jump up ^ By analogy with silver bullet.
References[edit]
Jump up ^ Richard W. Brislin (1980). "Cross-Cultural Research Methods: Strategies, Problems, Applications". In Irwin Altman; Amos Rapoport; Joachim F. Wohlwill. Environment and Culture. Springer. p. 73. ISBN 978-0-306-40367-5.
^ Jump up to: a b Abraham H. Maslow (1966). The Psychology of Science. p. 15.
Jump up ^ Bruce Klatt (1999). The ultimate training workshop handbook. McGraw-Hill Professional. p. 4. ISBN 978-0-07-038201-5.
Jump up ^ Timothy J. Cartwright (1990). The management of human settlements in developing countries: case studies in the application of microcomputers. Taylor & Francis. p. 230. ISBN 978-0-415-03124-0.
Jump up ^ Winther, Rasmus Grønfeldt (2014). James and Dewey on Abstraction. The Pluralist 9 (2), p. 20 http://philpapers.org/archive/WINJAD.pdf
Jump up ^ Thomas J. McQuade (2006). "Science and Markets as Adaptive Classifying Systems". In Elisabeth Krecké; Carine Krecké; Roger Koppl. Cognition and Economics. Emerald Group Publishing. p. 77. ISBN 978-0-7623-1378-5.
Jump up ^ Fortunes files - freebsd
Jump up ^ Lauby, Sharlyn. "The Law of the Instrument – Friday Distraction". HR Bartender. HR Bartender. Retrieved 19 April 2017.
Jump up ^ Green, Jonathon (1998). Dictionary of Slang. Cassell.
^ Jump up to: a b c d e "If Your Only Tool Is a Hammer Then Every Problem Looks Like a Nail". Quote Investigator. Quote Investigator. Retrieved 18 April 2017.
Jump up ^ Abraham Kaplan (1964). The Conduct of Inquiry: Methodology for Behavioral Science. San Francisco: Chandler Publishing Co. p. 28.
Jump up ^ Kagan, Robert (January 2004). Of Paradise and Power: America and Europe in the New World Order. Vintage Books. Retrieved 8 May 2017.
Jump up ^ William J. Brown; Raphael C. Malveau; Hays W. "Skip" McCormick; Thomas J. Mowbray (1998). AntiPatterns: Refactoring Software, Architectures, and Projects in Crisis. Wiley. p. 111. ISBN 978-0-471-19713-3.
Jump up ^ Gilgado, José. "Avoiding the law of the instrument". Jose M Dev. Jose M Dev. Retrieved 18 April 2017.
Jump up ^ "The "Law of the instrument"". The Economy of Meaning. The Economy of Meaning. Retrieved 18 April 2017.
Jump up ^ "Managing the complexity of human/machine interactions in computerized learning environments: guiding students' command process through instrumental orchestrations". International Journal of Computers for Mathematical Learning. 9: 281–307. 2004.
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