Trigonometry/Circles and Triangles/The Excircles

An excircle of a triangle is a circle that has as tangents one side of the triangle and the other two sides extended. There are three such circles, one corresponding to each side of the triangle.

The centre of each such circle, an excentre of the triangle, is at the intersection of the bisector of the angle opposite the side tangent to the circle and the external bisectors of the other two angles of the triangle. The proof is similar to that for the location of the incentre.

Let the sides of a triangle be a, b and c and the angles opposite be A, B and C respectively. Denote the area of the triangle by Δ. Write s = ¹⁄₂(a+b+c).

If the radius of the excircle touching side a is r_a, then $r_{a}={\frac {\Delta }{s-a}}$, with similar expressions for the other two excircles. If r is the inradius, we have

${\frac {1}{r_{a}}}+{\frac {1}{r_{b}}}+{\frac {1}{r_{c}}}={\frac {s-a}{\Delta }}+{\frac {s-b}{\Delta }}+{\frac {s-c}{\Delta }}={\frac {s}{\Delta }}={\frac {1}{r}}$

If the vertices of a triangle are ABC and the excircle touching side BC does so at point D, then AB+BD = AC+CD. (This is easily proved using the theorem that the two tangents from a point to a circle are equal in length.) Both these expressions are s, the semiperimeter.

The square of the distance of the excentre corresponding to side a from the circumcentre is R(R+2r_a), with similar expressions for the other two excentres. Also,

$\displaystyle r_{a}+r_{b}+r_{c}=r+4R.$

The area of the triangle is

$\displaystyle {\sqrt {r_{a}r_{b}r_{c}r}}.$

Three other expressions for r_a are

$r_{a}=a{{\cos({\frac {B}{2}})\cos({\frac {C}{2}})} \over {\cos({\frac {A}{2}})}}$

$r_{a}=4R\sin({\frac {A}{2}})\cos({\frac {B}{2}})\cos({\frac {C}{2}})$

$r_{a}=s\tan({\frac {A}{2}})$

with similar expressions for r_b and r_c.

It can readily be proved from the second of these expressions that if r_a = r + r_b + r_c then A is a right angle.

The distance of I_a from the three vertices A, B, C are

$4R\cos \left({\frac {B}{2}}\right)\cos \left({\frac {C}{2}}\right){\text{, }}4R\sin \left({\frac {A}{2}}\right)\cos \left({\frac {C}{2}}\right){\text{, }}4R\sin \left({\frac {A}{2}}\right)\cos \left({\frac {B}{2}}\right)$

respectively, with similar expressions for the other excentres.

$\displaystyle r.II_{a}.II_{b}.II_{c}=4R.IA.IB.IC$