INCIRCLE AND EXCIRCLES OF A TRIANGLE

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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 called the triangle's 'incenter'.
An 'excircle' or 'escribed circle' 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.

Contents

Explanation

Nine-point circle and Feuerbach point

Gergonne triangle and point

Coordinates of the incenter

Equations for four circles

External links

References

Explanation

The center of the incircle can be found as the intersection of the three internal angle bisectors.
The center of an excircle is the intersection of the internal bisector of one angle and the external bisectors of the other two. From this, it follows that the center of the incircle together with the three excircle centers form an orthocentric system.
The radii of the in- and excircles are closely related to the area of the triangle. If ''S'' is the triangle's area and its sides are ''a'', ''b'' and ''c'', then the radius of the incircle (also known as the 'inradius') is , the excircle at side ''a'' has radius , the excircle at side ''b'' has radius and the excircle at side ''c'' has radius . From these formulas we see in particular that the excircles are always larger than the incircle, and that the largest excircle is the one attached to the longest side.

Nine-point circle and Feuerbach point

The triangle's nine-point circle is tangent to the three excircles as well as to the incircle. The point where the nine-point circle touches the incircle is the Feuerbach point.

Gergonne triangle and point

The Gergonne point of a triangle is the symmedian point of its contact triangle.
Denoting the three vertices of the triangle by ''A'', ''B'' and ''C'' and the three points where the incircle touches the triangle by ''T_A'', ''T_B'' and ''T_C'' (where ''T_A'' is opposite of ''A'', etc.), the triangle ''T_AT_BT_C'' is known as the 'contact triangle' or 'Gergonne triangle' of ''ABC''. The incircle of ''ABC'' is the circumcircle of ''T_AT_BT_C''. The three lines ''AT_A'', ''BT_B'' and ''CT_C'' intersect in a single point, the triangle's 'Gergonne point' ''G''.


The contact triangle is also called the 'intouch triangle', and the touchpoints of the excircle with segments ''BC,CA,AC'' are the vertices of the 'extouch triangle'. The Gergonne triangle is also calld the 'excentral triangle', and 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 intouch triangle are given by

★ ''A-''vertex ''= 0'' : sec^''2''''(B/2)'' : sec^''2''''(C/2)''

★ ''B-''vertex '' = ''sec^''2''''(A/2)'' : ''0'' : sec^''2''''(C/2)''

★ ''C-''vertex '' = ''sec^''2''''(A/2)'' : sec^''2''''(B/2)'' : ''0''
Trilinear coordinates for the vertices of the extouch triangle are given by

★ ''A-''vertex ''= 0'' : csc^''2''''(B/2)'' : csc^''2''''(C/2)''

★ ''B-''vertex '' = ''csc^''2''''(A/2)'' : ''0'' : csc^''2''''(C/2)''

★ ''C-''vertex '' = ''csc^''2''''(A/2)'' : csc^''2''''(B/2)'' : ''0''
Trilinear coordinates for the vertices of the incentral triangle are given by

★ ''A-''vertex ''= 0 : 1 : 1''

★ ''B-''vertex ''= 1 : 0 : 1''

★ ''C-''vertex ''= 1 : 1 : 0''
Trilinear coordinates for the vertices of the excentral triangle are given by

★ ''A-''vertex ''= -1 : 1 : 1''

★ ''B-''vertex ''= 1 : -1 : 1''

★ ''C-''vertex ''= 1 : 1 : -1''
Trilinear coordinates for the Gergonne point are
sec²(''A''/2) : sec²(''B''/2) : sec²(''C''/2),
or, equivalently, ''bc''/(''b'' + ''c'' − ''a'') : ''ca''/(''c'' + ''a'' − ''b'') : ''ab''/(''a'' + ''b'' − ''c'').

Coordinates of the incenter

The Cartesian coordinates of the incenter are a weighted average of the coordinates of the three vertices. (The weights are positive so the incenter lies inside the triangle as stated above.) If the three vertices are located at $(x\_a,y\_a)$, $(x\_b,y\_b)$, and $(x\_c,y\_c)$, and the opposite sides of the triangle have lengths $a$, $b$, and $c$, then the incenter is at
:.
:
★ Trilinear coordinates for the incenter are 1 : 1 : 1.
:
★ Barycentric coordinates for the incenter are ''a'' : ''b'' : ''c''.

Equations for four circles

Let x : y : z be a variable point in trilinear coordinates, and let u = cos^''2''''(A/2)'', v = cos^''2''''(B/2)'', w = cos^''2''''(C/2)''. The four circles described above are given by these equations:
:
★ Incircle: ''u''^''2''''x''^''2'''' + v''^''2''''y''^''2'''' + w''^''2''''z''^''2'''' - 2vwyz - 2wuzx - 2uvxy = 0''
:
★ ''A-''excircle: ''u''^''2''''x''^''2'''' + v''^''2''''y''^''2'''' + w''^''2''''z''^''2'''' - 2vwyz + 2wuzx + 2uvxy = 0''
:
★ ''B-''excircle: ''u''^''2''''x''^''2'''' + v''^''2''''y''^''2'''' + w''^''2''''z''^''2'''' + 2vwyz - 2wuzx + 2uvxy = 0''
:
★ ''C-''excircle: ''u''^''2''''x''^''2'''' + v''^''2''''y''^''2'''' + w''^''2''''z''^''2'''' + 2vwyz + 2wuzx - 2uvxy = 0''

External links

★ Triangle centers by Antonio Gutierrez from Geometry Step by Step from the Land of the Incas.

★ Transitivity in Action — Remarkable Points in a Triangle at cut-the-knot

★ Incenters in Cyclic Quadrilateral at cut-the-knot

★ Equal Incircles Theorem at cut-the-knot

★ Five Incircles Theorem at cut-the-knot

★ Pairs of Incircles in a Quadrilateral at cut-the-knot

★ 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

★ Incircles

★ An interactive Java applet for the incenter

References

★ Clark Kimberling, "Triangle Centers and Central Triangles," ''Congressus Numerantium'' 129 (1998) i-xxv and 1-295.

★ Sándor Kiss, "The Orthic-of-Intouch and Intouch-of-Orthic Triangles," ''Forum Geometricorum'' 6 (2006) 171-177.