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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 2 of 7: Locate vertex CC
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∠A0CB0=∠Z\angle A_0CB_0=\angle Z
Detailed analysis

For every admissible ABCABC, the rays CACA and CBCB pass through B0B_0 and A0A_0. Hence ∠A0CB0=∠ACB=∠Z\angle A_0CB_0=\angle ACB=\angle Z, so CC lies on the fixed circle through A0,B0,ZA_0,B_0,Z.