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 5 of 7: Express the chord length
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AB=2∣proj⁡AB(OAOB)∣AB=2\left|\operatorname{proj}_{AB}(O_AO_B)\right|
Detailed analysis

The perpendicular projection of a circle center onto a chord is the chord midpoint. Therefore the projections of OA,OBO_A,O_B onto ABAB are the midpoints of C0A,C0BC_0A,C_0B, and their separation is AB/2AB/2.