Introduction : In this write-up, I will investigate a pedal point and a pedal triangle.

*Pedal point and Pedal triangles*

First, I will create a script for the general
construction of a pedal triangle to triangle ABC where P is any point in

the plane of ABC.

If you want to check the script, please click
here.

If pedal point P is the centroid(G) of triangle ABC, then the pedal
triangle looks like

If pedal point P is the incenter( I ) of triangle ABC, then the pedal
triangle looks like

If pedal point P is the orthocenter(H) of triangle ABC, then the pedal
triangle looks like

If pedal point P is the circumcenter(C) of triangle ABC, then the pedal triangle looks like

If you want to check GSP, click here.

If pedal point P is on a side of the triangle, then pedal point P equals
to R , S, and T and

the pedal triangles look like

If pedal point P is one of the vertices of triangles ABC, the pedal triangle looks like

When pedal point p is A, the points of A, R, T , and P are the same.

So, the pedal triangle become a line.

Similarly, if pedal point p is B or C, then the pedal triangle become
the line passing through B or C respectively.

*Simson Line*

All conditions in which the three vertices of the Pedal triangles are colinear are that pedal point P is on the cicumcircle of the triangle ABC.

If you want to check this fact by GSP, please click here

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