# 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_{a}I_{b}I_{c} is the **excentral triangle** of ABC.

A lies on the line I_{b}I_{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_{b}I_{c}I 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[edit]

The angles of this triangle are .

Its sides are **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle 4R\cos\left(\frac{A}{2}\right), 4R\cos\left(\frac{B}{2}\right), 4R\cos\left(\frac{C}{2}\right)}**
.

Its area is

where Δ is the area of the original triangle.

Its circumradius is 2R.