Pedal triangle

In plane 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 and $P$ is the orthocenter, then the angles of $△ LMN$ are $180° - 2 A$ , $180° - 2 B$ and $180° - 2 C$ .^[1]

The location of the chosen point $P$ relative to the chosen triangle $△ ABC$ gives rise to some special cases:

If $P$ is the orthocenter, then $△ LMN$ is the orthic triangle.
If $P$ is the incenter, then $△ LMN$ is the intouch triangle.
If $P$ is the circumcenter, then $△ LMN$ is the medial triangle.
If $P$ is on the circumcircle of the triangle, $△ LMN$ collapses to a line (the pedal line or Simson line).

The vertices of the pedal triangle of an interior point $P$ , as shown in the top diagram, divide the sides of the original triangle in such a way as to satisfy Carnot's theorem:^[2]

$|AN|^{2}+|BL|^{2}+|CM|^{2}=|NB|^{2}+|LC|^{2}+|MA|^{2}.$

^ "Trigonometry/Circles and Triangles/The Pedal Triangle - Wikibooks, open books for an open world". en.wikibooks.org. Retrieved 2020-10-31.
^ Alfred S. Posamentier; Charles T. Salkind (1996). Challenging problems in geometry. New York: Dover. pp. 85-86. ISBN 9780486134864. OCLC 829151719.

[1] "Trigonometry/Circles and Triangles/The Pedal Triangle - Wikibooks, open books for an open world". en.wikibooks.org. Retrieved 2020-10-31.

[2] Alfred S. Posamentier; Charles T. Salkind (1996). Challenging problems in geometry. New York: Dover. pp. 85-86. ISBN 9780486134864. OCLC 829151719.

[1]

[2]