Assignment 9: Pedal Triangle

For this exploration, we will take a pedal triangle and animate the point P about the incircle of triangle ABC. While point P is animated, we will observe the traces of the midpoints of the pedal triangle.

Here shows the traces of the midpoints in pink. The paths of these midpoints appear to produce ellipses.

For an animation, here is a GSP file to watch or play around with!

Now let's observe the trace of the midpoints when triangle ABC is a right triangle...

Here we see that one of the midpoints has traced the path of a circle!

This is no coincidence though. Check out the animation with the perpendicular lines not hidden here. Notice that the quadrilateral created by points P, R, A, and T is consistently a rectangle. This may be more apparent in this GSP file.

I bet that if triangle ABC were a right isosceles triangle, we would have a circle with two congruent ellipses. Let's find out...

Indeed! The two ellipses appear congruent. It also appears that there may be a connection between the ellipses and the incenter. Possible further exploration could be to decipher this connection and prove it!

How about if triangle ABC was an equilateral triangle!

This produces a very appealing, symmetric image! It appears the ellipse rotates 120 degrees about the incenter.

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