MathDB

Problem 3

Part of 2024 Nigerian MO Round 3

Problems(1)

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

Source: 2024 Nigerian MO Senior Round 3 P3

4/28/2024
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.
geometrycircumcircletrigonometry