Assignment 9:Pedal Triangles

From Jim Wilson’s assignment page:  1a. 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.

Sketch

9. Find all conditions in which the three vertices of the Pedal triangle are colinear (that is, it is a degenerate triangle). This line is called the Simson Line.

By exploration, it can be discovered that the pedal triangle becomes degenerate when P lies on a vertex of the triangle.  The placing P on the circumcircle might be a good bet for finding all such places.  (This can be easily done using the merge point feature in the edit menu).

Locate the midpoints of the sides of the Pedal Triangle. Construct a circle with center at the circumcenter of triangle ABC such that the radius is larger than the radius of the circumcircle. Trace the locus of the midpoints of the sides of the Pedal Triangle as the Pedal Point P is animated around the circle you have constructed. What are the three paths?

The three paths appear to be ellipses.  Very interesting construction and fun to animate.

Repeat where the path is the circumcircle.

In this case the paths traced by the midpoints still appear to be ellipses, but these each ellipse passes through one vertex of the original triangle.

In particular, find the envelope of the Simson line as the Pedal point is moved along the circumcircle.

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