Incircle and excircles of a triangle

Incircle and incenter

Suppose △ A B C 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 be the touchpoints where the incircle touches BC , AC and AB

Incenter

The incenter is the point where the internal angle bisectors of ∠ A B C , ∠ B C A ,  and  ∠ B A C 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 )

Trilinear coordinates of the incenter

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

  1 : 1 : 1.

Barycentric coordinates of the incenter

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

where a , b , and c are the lengths of the sides of the triangle, or equivalently (using the law of sines) by

sin ⁡ ( A ) : sin ⁡ ( B ) : sin ⁡ ( C )

where A , B , and C are the angles at the three vertices.

Cartesian coordinates of the incenter

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 ) , ( x b , y b ) , and ( x c , y c ) , and the sides opposite these vertices have corresponding lengths a , b , and 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 .

Distance of the incenter to the vertices

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

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.

Additionally,

I A ⋅ I B ⋅ I C = 4 R r 2 ,

where R and r are the triangle's circumradius and inradius respectively.

Other incenter properties

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.

Distances between vertex and nearest touchpoints

The distance from a vertex to the two nearest touchpoints are equaland after some elementary algebra using you will find:

d ( A , T B ) = d ( A , T C ) ) = 1 2 ( b + c − a )

Other incircle properties

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

r = x y z x + y + z

and the area of the triangle is

Δ = x y z ( x + y + z ) .

If the altitudes from sides of lengths a, b, and c are h_a, h_b, and h_c 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 .

The product of the incircle radius r and the circumcircle radius R of a triangle with sides a, b, and c is

r R = a b c 2 ( a + b + c ) .

Some relations among the sides, incircle radius, and circumcircle radius are:

a b + b c + c a = s 2 + ( 4 R + r ) r , a 2 + b 2 + c 2 = 2 s 2 − 2 ( 4 R + 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.

Denoting the center of the incircle of triangle ABC as I, we have

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

and

I A ⋅ I B ⋅ I C = 4 R r 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. Thus for example for vertex B and adjacent tangencies T_A and T_C,

B T A = B T C = B C + A B − A C 2 .

The incircle radius is no greater than one-ninth the sum of the altitudes.

The squared distance from the incenter I to the circumcenter O is given by

O I 2 = R ( R − 2 r ) ,

and the distance from the incenter to the center N of the nine point circle is

I N = 1 2 ( R − 2 r ) < 1 2 R .

The incenter lies in the medial triangle (whose vertices are the midpoints of the sides).

Relation to area of the triangle

The radii of the incircle is related to the area of the triangle. The ratio of the area of the incircle to the area of the triangle is less than or equal to π 3 3 , with equality holding only for equilateral triangles.

Suppose △ A B C 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 is right. Thus the radius T_cI is an altitude of △ I A B . Therefore, △ I A B has base length c and height r, and so has area 1 2 c r . Similarly, △ I A C has area 1 2 b r and △ I B C has area 1 2 a r . Since these three triangles decompose △ A B C , we see that the area Δ  of  △ A B C is:

Δ = 1 2 ( a + b + c ) r = s r ,      and      r = Δ s ,

where Δ is the area of △ A B C and s = 1 2 ( a + b + c ) is its semiperimeter.

For an alternative formula, consider △ I T c A . This is a right-angled triangle with one side equal to r and the other side equal to r cot ⁡ ∠ A 2 . The same is true for △ I B ′ 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 )

Gergonne triangle and point

The Gergonne triangle (of ABC) is defined by the 3 touchpoints of the incircle on the 3 sides. The touchpoint opposite A is denoted T_A, etc.

This Gergonne triangle T_AT_BT_C 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

where K , r , s are the area, radius of the incircle and semiperimeter of the original triangle, and a , b , c are the side lengths of the original triangle. This is the same area as that of the extouch triangle.

The three lines AT_A, BT_B and CT_C 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.

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.

Trilinear coordinates for the vertices of the intouch triangle are given by

vertex A = 0 : sec 2 ⁡ ( B 2 ) : sec 2 ⁡ ( C 2 )

vertex B = sec 2 ⁡ ( A 2 ) : 0 : sec 2 ⁡ ( C 2 )

vertex C = sec 2 ⁡ ( A 2 ) : sec 2 ⁡ ( B 2 ) : 0

Trilinear coordinates for the Gergonne point are given by

sec 2 ⁡ ( A 2 ) : sec 2 ⁡ ( B 2 ) : sec 2 ⁡ ( C 2 ) ,

or, equivalently, by the Law of Sines,

b c b + c − a : c a c + a − b : a b a + b − c .

Excircles and Excenters

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.

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. 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.

Trilinear coordinates of excenters

While A triangle's incenter has trilinear coordinates 1 : 1 : 1. Its excenters (the centers of its excircles) have trilinears − 1 : 1 : 1 , 1 : − 1 : 1 and 1 : 1 : − 1.

Exradii

The radii of the excircles are called the exradii.

the exradius of the Excircle opposite A (so touching BC, centered at J A is

r a = s ( s − b ) ( s − c ) s − a .

where s = 1 2 ( a + b + c )

Derivation of exradii formula

Let the excircle at side AB touch at side AC extended at G, and let this excircle's radius be r c and its center be J c .

Then J c G is an altitude of △ A C J c , so △ A C J c has area 1 2 b r c . By a similar argument, △ B C J c has area 1 2 a r c and △ A B J c has area 1 2 c r c . Thus the area of triangle △ A B C ( Δ ) is

Δ = 1 2 ( a + b − c ) r c = ( s − c ) r c .

So, by symmetry, where r i is the radius of the incircle

Δ = s r i = ( 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

Combining this with the identity 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

But Δ = 1 2 b c 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 ) ,

which is Heron's formula.

Combining this with s r i = Δ , we have

r i 2 = Δ 2 s 2 = ( s − a ) ( s − b ) ( s − c ) s .

Similarly, ( s − a ) r a = Δ gives

r a 2 = s ( s − b ) ( s − c ) s − a

and

r a = s ( s − b ) ( s − c ) s − a .

Other exradii properties

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:

Δ = r i r 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 , with equality holding only for equilateral triangles.

Other excircle properties

The circular hull of the excircles is internally tangent to each of the excircles, and thus is an Apollonius circle. The radius of this Apollonius circle is r 2 + s 2 4 r where r is the incircle radius and s is the semiperimeter of the triangle.

The following relations hold among the inradius r, the circumradius R, the semiperimeter s, and the excircle radii r_a, r_b, r_c:

r a + r b + r c = 4 R + r , 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 ,

The circle through the centers of the three excircles has radius 2R.

If H is the orthocenter of triangle ABC, then

r a + r b + r c + r = A H + B H + C H + 2 R , r a 2 + r b 2 + r c 2 + r 2 = A H 2 + B H 2 + C H 2 + ( 2 R ) 2 .

Nagel triangle and Nagelpoint

The Nagel triangle or extouch triangle of ABC is denoted by the vertices T_A, T_B and T_C that are the three points where the excircles touch the reference triangle ABC and where T_A is opposite of A, etc. This triangle T_AT_BT_C is also known as the extouch triangle of ABC. The circumcircle of the extouch triangle T_AT_BT_C is called the Mandart circle.

The three lines AT_A, BT_B and CT_C 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 )

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 A = 0 : csc 2 ⁡ ( B 2 ) : csc 2 ⁡ ( C 2 )

vertex B = csc 2 ⁡ ( A 2 ) : 0 : csc 2 ⁡ ( C 2 )

vertex C = csc 2 ⁡ ( A 2 ) : csc 2 ⁡ ( B 2 ) : 0

Trilinear coordinates for the Nagel point are given by

csc 2 ⁡ ( A 2 ) : csc 2 ⁡ ( B 2 ) : csc 2 ⁡ ( C 2 ) ,

or, equivalently, by the Law of Sines,

b + c − a a : c + a − b b : a + b − c c .

It is the isotomic conjugate of the Gergonne point.

Relation to area of the triangle

The radii of the incircles and excircles are closely related to the area of the triangle.

Incircle

Suppose △ A B C 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 is right. Thus the radius C'I is an altitude of △ I A B . Therefore, △ I A B has base length c and height r, and so has area 1 2 c r . Similarly, △ I A C has area 1 2 b r and △ I B C has area 1 2 a r . Since these three triangles decompose △ A B C , we see that

Δ = 1 2 ( a + b + c ) r = s r ,      and      r = Δ s ,

where Δ is the area of △ A B C and s = 1 2 ( a + b + c ) is its semiperimeter.

For an alternative formula, consider △ I C ′ A . This is a right-angled triangle with one side equal to r and the other side equal to r cot ⁡ ∠ A 2 . The same is true for △ I B ′ 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 )

Excircles

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 and its center be I c . Then I c G is an altitude of △ A C I c , so △ A C I c has area 1 2 b r c . By a similar argument, △ B C I c has area 1 2 a r c and △ A B I c has area 1 2 c r c . Thus

Δ = 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 .

By the Law of Cosines, we have

cos ⁡ A = b 2 + c 2 − a 2 2 b c

Combining this with the identity 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

But Δ = 1 2 b c 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 ) ,

which is Heron's formula.

Combining this with s r = Δ , we have

r 2 = Δ 2 s 2 = ( s − a ) ( s − b ) ( s − c ) s .

Similarly, ( s − a ) r a = Δ gives

r a 2 = s ( s − b ) ( s − c ) s − a

and

r a = s ( s − b ) ( s − c ) s − a .

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:

Δ = r r 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 , with equality holding only for equilateral triangles.

Nine-point circle and Feuerbach point

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).

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.

Gergonne triangle and point

The Gergonne triangle (of ABC) is defined by the 3 touchpoints of the incircle on the 3 sides. The touchpoint opposite A is denoted T_A, etc.

This Gergonne triangle T_AT_BT_C 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

where K , r , s are the area, radius of the incircle and semiperimeter of the original triangle, and a , b , c are the side lengths of the original triangle. This is the same area as that of the extouch triangle.

The three lines AT_A, BT_B and CT_C 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.

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.

Trilinear coordinates for the vertices of the intouch triangle are given by

vertex A = 0 : sec 2 ⁡ ( B 2 ) : sec 2 ⁡ ( C 2 )

vertex B = sec 2 ⁡ ( A 2 ) : 0 : sec 2 ⁡ ( C 2 )

vertex C = sec 2 ⁡ ( A 2 ) : sec 2 ⁡ ( B 2 ) : 0

Trilinear coordinates for the Gergonne point are given by

sec 2 ⁡ ( A 2 ) : sec 2 ⁡ ( B 2 ) : sec 2 ⁡ ( C 2 ) ,

or, equivalently, by the Law of Sines,

b c b + c − a : c a c + a − b : a b a + b − c .

Nagel triangle and point

The Nagel triangle of ABC is denoted by the vertices X_A, X_B and X_C that are the three points where the excircles touch the reference triangle ABC and where X_A is opposite of A, etc. This triangle X_AX_BX_C is also known as the extouch triangle of ABC. The circumcircle of the extouch triangle X_AX_BX_C is called the Mandart circle. The three lines AX_A, BX_B and CX_C are called the splitters of the triangle; they each bisect the perimeter of the triangle, and they 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 A = 0 : csc 2 ⁡ ( B 2 ) : csc 2 ⁡ ( C 2 )

vertex B = csc 2 ⁡ ( A 2 ) : 0 : csc 2 ⁡ ( C 2 )

vertex C = csc 2 ⁡ ( A 2 ) : csc 2 ⁡ ( B 2 ) : 0

Trilinear coordinates for the Nagel point are given by

csc 2 ⁡ ( A 2 ) : csc 2 ⁡ ( B 2 ) : csc 2 ⁡ ( C 2 ) ,

or, equivalently, by the Law of Sines,

b + c − a a : c + a − b b : a + b − c c .

It is the isotomic conjugate of the Gergonne point.

Incentral and excentral triangles

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

  ( vertex opposite B ) = 1 : 0 : 1

  ( 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

( vertex opposite B ) = 1 : − 1 : 1

( vertex opposite C ) = 1 : 1 : − 1

Equations for four circles

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:

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 ± x cos ⁡ A 2 ± y cos ⁡ B 2 ± z cos ⁡ 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 ± − x cos ⁡ A 2 ± y cos ⁡ B 2 ± z cos ⁡ 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 ± x cos ⁡ A 2 ± − y cos ⁡ B 2 ± z cos ⁡ 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 ± x cos ⁡ A 2 ± y cos ⁡ B 2 ± − z cos ⁡ C 2 = 0

Euler's theorem

Euler's theorem states that in a triangle:

( R − r i n ) 2 = d 2 + r i n 2 ,

where R and r_in 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 2 + r e x 2 ,

where r_ex is the radius of one of the excircles, and d is the distance between the circumcenter and this excircle's center.

Generalization to other polygons

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.