Given a triangle and a point lying on a circle concentric with the circumcircle but with twice the area, we create another triangle by reflecting the point in each side of the triangle. We first show that the new triangle is the same area as the old one, then we examine the envelope of the sides of the new triangle as the point traverses the circle.

We put a general triangle in the special location so that its vertices have coordinates (0,0), (a,0), (b,c)

Inspection of the equation (another coffee break one) reveals that the envelope of the side is a conic. By drawing an ellipse and giving it the envelope’s equation, we can examine the coordinates of the foci. One is at the origin, the other, we see, is at the point (b, b(a-b)/c).

The envelopes of all three sides share a focus at the orthocenter.

Name	Input
Orthocenter
Area Eq
Focus Eq