MathLabs

Problem 3

Let ABCABC be a triangle. The bisector of angle AA meets segment BCBC at XX and the circumcircle at YY. Let rA=AX/AYr_A=AX/AY, and define rB,rCr_B,r_C similarly. Prove that rA/sin⁡2A+rB/sin⁡2B+rC/sin⁡2C≥3r_A/\sin^2A+r_B/\sin^2B+r_C/\sin^2C\ge3, with equality if and only if the triangle is equilateral.
Step 4 of 5: Use the sine bound
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1sin⁡2t≥1\frac1{\sin^2t}\ge1
Detailed analysis

Every relevant angle lies between 00 and π\pi, so sin⁡2t≤1\sin^2t\le1 and 1/sin⁡2t≥11/\sin^2t\ge1. Therefore the target sum is at least sA+sB+sC≥3s_A+s_B+s_C\ge3.