Two regular hexagons connected by a rectangle. Prove that the five vertices lie on a circle.
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Solution
First draw the line segments GC. and CI. By adding the angles on the left side of these segments, we see they add up to 180. So GI is a straight line. Now triangle GIJ is a right triangle with hypothenuse GJ, and from the reversed Thales theorem it follows that I lies on the circle with GJ as diameter.
In a similar fashion triangles GHJ and GAJ are shown to be right triangles with hypothenuse GJ and so all points G, H, A, I and J lie on the same circle.