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.
If P is on the circumcircle of the triangle, LMN collapses to a line. This is then called the pedal line, or sometimes the Simson line after Robert Simson. The vertices of the pedal triangle of an interior poi
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