Written by Sunny Yoon

Let triangle ABC be any triangle. Then if P is any point in the plane, then the triangle formed by constructing perpendiculars to the sides of ABC (extended if necessary) locate three points R, S, and T that are the intersections. Triangle RST is the Pedal Triangle for Pedal Point P

For example, when P is outside of triangle ABC.

When P is inside triangle ABC

What if pedal point P is the centroid of triangle ABC?

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What if pedal point P is the incenter of triangle ABC?

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What if pedal point P is the orthocenter of triangle ABC?

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What if pedal point P is the circumcenter of triangle ABC?

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Locate the midpoints of the sides of the pedal triangle. Construct a circumcircle of triangle ABC. Trace the locus of the midpoints of the sides of the pedal triangle as the pedal point P is animated around the circle you constructed. What are the 3 paths?

In particular, find the envelope of the Simson line as the Pedal point is moved along the circumcircle. Note, you will need to trace the image of the line, not the segment.