Given a triangle, find three circles that are tangent to the circumcircle of the triangle at the three points and that are also tangent to each other.
Construct three squares with two vertices on a side and the remaining two vertices on each remaining side. The squares are readily found by drawing the altitudes and using similarity of the triangles.
The three pairs of square vertices form three triangles with the triangle vertices. The circumcircles of the triangles are the required circles.
THINGS TO TRY
Was Édouard Lucas (1842–1891) the inventor of this gimmick? The problem evokes Descartes's theorem or Soddy circles. Who would think of using these easy-to-construct squares, which apparently do not seem to be related to the problem?
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