MathLabs

Problem 5

Let ABC be a triangle and X an interior point of ABC. Show that at least one of the angles XAB, XBC, XCA is less than or equal to 30 degrees.
Step 2 of 4: Obtain the product identity and monotonicity
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∏cycsin⁡(A−α)sin⁡α=1,fA(t)=sin⁡(A−t)sin⁡t=sin⁡Acot⁡t−cos⁡A\prod_{cyc}\frac{\sin(A-\alpha)}{\sin\alpha}=1,\qquad f_A(t)=\frac{\sin(A-t)}{\sin t}=\sin A\cot t-\cos A
Detailed analysis

Multiplying the three distance identities and cancelling AX·BX·CX yields the product of the three ratios sin(A−alpha)/sin(alpha), cyclically, equal to 1. For fixed A, f_A(t)=sin(A−t)/sin(t)=sin(A)cot(t)−cos(A) is strictly decreasing wherever the angles are valid.