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ABC is equilateral if circles (XYZ) and (ABC) are equal

Source: 2009 Cuba 2.72

August 27, 2024
Equilateralincenterequal circlesgeometry

Problem Statement

Let II be the incenter of an acute riangle ABCABC. Let CA(A,AI)C_A(A, AI) be the circle with center AA and radius AIAI. Circles CB(B,BI)C_B(B, BI), CC(C,CI)C_C(C, CI) are defined in an analogous way. Let X,Y,ZX, Y, Z be the intersection points of CBC_B with CCC_C, CCC_C with CAC_A, CAC_A with CBC_B respectively (different than II) . Show that if the radius of the circle that passes through the points X,Y,ZX, Y, Z is equal to the radius of the circle that passes through points AA, BB and CC then triangle ABCABC is equilateral.
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