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 5 of 5: Check equality
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A=B=C=60∘A=B=C=60^\circ
Detailed analysis

Equality requires all three AM-GM factors and all three sine bounds to be equalities. The equations B+A/2=C+B/2=A+C/2=90∘B+A/2=C+B/2=A+C/2=90^\circ force A=B=C=60∘A=B=C=60^\circ, and an equilateral triangle indeed gives equality.