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Problem 4

Let A0B0C0A_0B_0C_0 and A1B1C1A_1B_1C_1 be any two acute-angled triangles. Consider all triangles ABCABC similar to A1B1C1A_1B_1C_1 (with corresponding vertices) and circumscribed about A0B0C0A_0B_0C_0, where A0A_0 lies on BCBC, B0B_0 on CACA, and C0C_0 on ABAB. Determine and construct the triangle of maximum area.
Step 4 of 7: Use the two circles meeting at C0C_0
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ΓA=⊙(B0C0X),ΓB=⊙(C0A0Y)\Gamma_A=\odot(B_0C_0X),\quad\Gamma_B=\odot(C_0A_0Y)
Detailed analysis

Let their centers be OA,OBO_A,O_B. The variable side ABAB always passes through C0C_0 and cuts ΓA,ΓB\Gamma_A,\Gamma_B again at A,BA,B, respectively.