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Concept# Pedal triangle

Summary

In geometry, a pedal triangle is obtained by projecting a point onto the sides of a triangle.
More specifically, consider a triangle ABC, and a point P that is not one of the vertices A, B, C. Drop perpendiculars from P to the three sides of the triangle (these may need to be produced, i.e., extended). Label L, M, N the intersections of the lines from P with the sides BC, AC, AB. The pedal triangle is then LMN.
If ABC is not an obtuse triangle, P is the orthocenter then the angles of LMN are 180°−2A, 180°−2B and 180°−2C.
The location of the chosen point P relative to the chosen triangle ABC gives rise to some special cases:

- If P = orthocenter, then LMN = orthic triangle.
- If P = incenter, then LMN = intouch triangle.
- If P = circumcenter, then LMN = medial triangle.

Official source

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