A Circle Defined from an Equilateral Triangle via Pythagoras
Consider an equilateral triangle and the set of points such that . Then is either 30° or 150°, depending on whether is inside or outside , and so the locus of is a circle through and .
Use the "rotate" slider to rotate the triangle by . This gives:
Snapshot 1: with
Snapshot 2: the point is outside the triangle with
Snapshot 3: the point is at the top vertex of the triangle and is undefined
Suppose is inside the circle.
Rotate by so that falls along . Call the rotated triangle ; since and have equal lengths, . Also, since and have equal lengths and , is equilateral and and have equal lengths. Then:
Since and is equilateral, .
When is outside , a similar argument gives .
 AoPS Online. (Sep 10, 2019) artofproblemsolving.com/community/c3t48f3h1903996_geometry.