Begin with any triangle ABC.

Construct the altitudes of triangle ABC. Let A' be the intersection of the altitude from vertex A with opposite side BC. Label B' and C' similarly. Each of A', B', and C' is a foot an altitude of triangle ABC. Triangle A'B'C' is called the orthic triangle.

Construct the circumcircle of triangle ABC. The circumcircle is the circumscribed circle about triangle ABC. It passes through all three vertices of ABC. Its center is the circumcenter, the intersection of the perpendicular bisectors of the sides of AC.

Let A'' be the intersection (other than A) of the altitude from A with the circumcircle.

Label B'' and C'' similarly. We will call A''B''C'' the extended altitude triangle.

What is the relationship between the orthic (gray) triangle and the extended altitude (green) triangle?

We will begin by looking at our construction using a "nice triangle" and making some comparisions

Observations:

1. Area of Orthic Triangle = 1/4 Area of Extended Altitude Triangle

2. Perimeter of Orthic Triangle = 1/2 Area of Extended Altitude Triangle

3. Orthic Triangle is similar to Extended Altitude Triangle by a factor of 1/2.

4. Corresponding sides of Orthic and Extended Altitude Triangles are parallel.

5. Orthic Triangle is entirely inside of Extended Altitude Triangle.

We can move the vertices of the original triangle ABC to see if our 5 observations hold in less "nice" cases.

The triangle we looked at above is acute. Move vertex B to see what happens as the triangle becomes right and then obtuse. As long as the triangle is acute, all 5 observations are true. When ABC becomes right, the orthic and extended altitude triangles disappear as the feet of the altitudes on the legs of the triangle converge to the vertex of the right angle.

As angle ABC grows larger than 90, the orthic and extended altitude triangles reappear.

All observations are preserved except #5. The triangles still overlap but the orthic triangle is no longer entirely inside of the extended altitude triangle.

As angle ABC continues to grow the observations are preserved until the altitude from A become tangent to the circle at point A, that is A = A''. (A symmetrical situation occurs when C'' converges to C.)

As ABC grows beyond that point, all 5 observations appear to fail. C'' is "stuck " at C and A'' is "stuck" at A, so that the extended altitude triangle A''B''C'' do longer adjusts to maintain similarity and parallel sides with the orthic triangle A'B'C' which continues to change.

However, I finally realized that the problem was with the GSP sketch instead of the observations. Somehow A'' fails to move through A as it should once the tangency occurs then passes.

Construct appropriate intersections of the altitudes and circumcircle, resulting in the yellow triangle. Then the relationship between the yellow extended altitude triangle and the orthic triangle preserves all observations with acute triangles and observations 1-4 with obtuse triangles.