A clue fact is that points B, G and J are on a line. In fact, angle(GBC) is half the angle(GIC) = angle(A*AC). Hence BG must pass through the middle J of arc A*C. Then angle(FGB) is half of the angle(BIF)=angle(BIA), which is double the angle(JGM). Thus, F, G, M are on a line, completing the proof for the beautiful cross.

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