Written by Sunny Yoon

First I'm going to construct a triangle ABC. Then, I'll find the orthocenter of ABC and label it H. After that, I can construct three other triangles, HBC, HAC, HAB. Construct the orthocenter of those three triangles.

Finally, I'm going to construct circumcircles of all four triangles; ABC, HBC, HAC, and HAB.

Here is the picture.

Now I'm going to examine the triangles formed by the points where the extended altitudes meet the circumcircle. Here is the picture.

How is it related to the orthic triangle? It is similar to the orthic triangle and the ratio is 2 : 1.

Is it true if the original triangle is an obtuse triangle?

Click here for the GSP file.

A different investigation.

I constructed an acute triangle ABC and circumcircle. Then I constructed altitudes AD, BE, and CF. Label the intersection between the circumcircle and extended altitudes as P, Q, and R.

Click here for a GSP file.

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