Let the triangle be ABC, and the point P. Drop perpendiculars from P to the three sides of the triangle (these may need to be produced, ie extended). Label L, M, N the intersections of the lines from P with the sides BC, AC, AB. The pedal triangle is then LMN.
The location of the chosen point P relative to the chosen triangle ABC gives rise to some special cases:
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.
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