Assignment # 9

Pedal Triangles and the Simson Line

Whenever our pedal point is on the Circumcircle, we get the Simson Line:


Now, if we were to trace the Simson Line, we get the following set-up:


What happens when our Simson Line intersects the line that connects our P to our orthocenter?

Here, the result is a circle. By plotting the lotus of points that result from crossing our Pedal Triangle with our Orthocenter segment, we get the brown circle.

How are the two circles related? Well - I roughly put in what I thought to be the diameter, and it seems to be equal to the radius of our circumcircle. How does that relate our two circles? Let's take the midpoint of our "diameter" segment and look at the two resulting areas, etc:

As we can see, the size of our circumcircle is four times the size of the locus of the intersection.

Now, if we get two pedal points on our circumcircle and then look at their intersecting angle, comparing it to the angle of the arc that results from our intersection, we discover the following relationship:

Looking at different values made by those arcs, there doesn't seem to be any kind of a pattern to our construction.

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