MathLabs

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 7 of 7: Construct the maximum
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AB∥OAOBAB\parallel O_AO_B
Detailed analysis

Draw through C0C_0 the line parallel to OAOBO_AO_B; its second intersections with the two circles give AA and BB. The prescribed angles then make the resulting similar triangle the unique maximum (up to the admissible orientation). Since all admissible triangles are similar, maximizing ABAB maximizes area.