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# how to find sides and angles of pedal triangle??

Apoorva Arora IIT Roorkee
askIITians Faculty 181 Points
7 years ago

Given a point P inside the triangle ABC, The pedal triangle of P is the triangle whose polygon vertices are the feet of the perpendiculars from P to the sidelines. The pedal triangle of a triangle with trilinear coordinates$\alpha :\beta :\gamma$ and angles A B and C has trilinear vertex matrix
$\begin{bmatrix} 0 & \beta +\alpha cosC& \gamma +\alpha cosB\\ \alpha +\beta cosC& 0& \gamma +\beta cosA\\ \alpha +\gamma cosB & \beta +\gamma cosA& 0 \end{bmatrix}$
The side lengths are,
$a'=\frac{abc\sqrt{\beta ^{2}+\gamma ^{2}+2\beta \gamma cosA}}{2R|a\alpha +b\beta +c\gamma |}$
$b'=\frac{abc\sqrt{\alpha ^{2}+\gamma ^{2}+2\alpha \gamma cosB}}{2R|a\alpha +b\beta +c\gamma |}$
$c'=\frac{abc\sqrt{\alpha ^{2}+\beta ^{2}+2\alpha\beta \gamma cosC}}{2R|a\alpha +b\beta +c\gamma |}$
where R is the circumradius of the triangle ABC.
Thanks and Regards
Apoorva Arora
IIT Roorkee