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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 3 of 7: Locate the other vertices
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A∈⊙(B0C0X),B∈⊙(C0A0Y)A\in\odot(B_0C_0X),\quad B\in\odot(C_0A_0Y)
Detailed analysis

The same inscribed-angle argument gives AA on the circle through B0,C0,XB_0,C_0,X and BB on the circle through C0,A0,YC_0,A_0,Y. Thus the three vertex loci are fixed circles.