Pedal Triangles

By: Lacy Gainey



Click here for a script tool of a pedal triangle.
The tool will produce a pedal triangle similar to the one below.

ΔRST is the pedal triangle of ΔABC and the pedal point, P.
First, lets observe what happens when P lies on the side of ΔABC.

When we place P on side BC, R and P become the same point.  The same phenomenon occurs when we place P on the other two sides as well. 
When P lies on the side of ΔABC, P is one of the vertices of the pedal triangle.

What if P is one of the vertices of ΔABC?

We see that the pedal triangle disappears and we are left with a line segment.
All three vertices of the pedal triangle are collinear.  This line is referred to as the Simson line.

Are the vertices of ΔABC the only points where the Simson line appears?
Move P around, outside of ΔABC and see if you can find another point where the Simson line appears.  Keep an eye out for any patterns.

One of the points I found is illustrated below.

We know the Simson line appears when P is one of the vertices of ΔABC, so lets try moving P around the circumcircle of ΔABC.
This is illustrated in this GSP file.

When P lies on the circumcircle, the Simson line appears.  Additionally, as P rotates around the circumcircle, the Simson line rotates as well.

We can conclude that the Simson line is formed when the pedal point, P, lies on the circumcirlce of ΔABC.

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