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cos(a+b)/cos(a-b)

Source: 2024 Nigerian MO Senior Round 3 P3

April 28, 2024
geometrycircumcircletrigonometry

Problem Statement

Let ABCABC be a triangle, and let OO be its circumcenter. Let CO‾∩AB≡D\overline{CO}\cap AB\equiv D. Let ∠BAC=α\angle BAC=\alpha, and ∠CBA=β\angle CBA=\beta. Prove that ODOC=∣cos⁡(α+β)cos⁡(α−β)∣\dfrac{OD}{OC}=\Bigg|\dfrac{\cos(\alpha+\beta)}{\cos(\alpha-\beta)}\Bigg|\\
For clarification, CO‾\overline{CO} represents the line COCO, and ACAC represents the segment ACAC. Cases in which DD doesn't exist should be ignored.
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