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(Template:Frac) 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 $8R^2\cos \left(\frac{A}{2}\right)\cos \left(\frac{B}{2}\right)\cos \left(\frac{C}{2}\right)=2Rs=\left(\frac{1}{2}\right)\mathrm{\Delta }\text{cosec}\left(\frac{A}{2}\right)\text{cosec}\left(\frac{B}{2}\right)\text{cosec}\left(\frac{C}{2}\right)$

where Δ is the area of the original triangle.

Its circumradius is 2R.