Trigonometry/Circles and Triangles/The Excentral Triangle

Let ABC be a triangle, its incentre be I and its three excentres be I_a, I_b and I_c. Then I_aI_bI_c is the excentral triangle of ABC.

A lies on the line I_bI_c and is the foot of the perpendicular from I_a to that line, and similarly for B and C. Thus ABC is the pedal triangle (see later) of its excentral triangle. Further, these perpendiculars intersect at I, so I is the orthocentre of the excentral triangle.

I and I_a lie on the bisector of angle BAC and similarly for the other excentres.

The quadrilaterals IBCI_a, IACI_b and IABI_c are cyclic.

I_a is the orthocentre of the triangle I_bI_cI and similarly for the other excentres.

The distance II_a is 4Rsin(^A⁄₂) and similarly for the other excentres.

Properties of the Excentral Triangle

The angles of this triangle are $90^{0}-{\frac {A}{2}},90^{0}-{\frac {B}{2}}{\text{ and }}90^{0}-{\frac {C}{2}}$.

Its sides are $4R\cos \left({\frac {A}{2}}\right),4R\cos \left({\frac {B}{2}}\right),4R\cos \left({\frac {C}{2}}\right)$.

Its area is $8R^{2}\cos \left({\frac {A}{2}}\right)\cos \left({\frac {B}{2}}\right)\cos \left({\frac {C}{2}}\right)=2Rs=\left({\frac {1}{2}}\right)\Delta {\text{cosec}}\left({\frac {A}{2}}\right){\text{cosec}}\left({\frac {B}{2}}\right){\text{cosec}}\left({\frac {C}{2}}\right)$

where Δ is the area of the original triangle.

Its circumradius is 2R.