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 3 of 4: Assume all three marked angles exceed 30 degrees
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α,β,γ>30∘ ⟹ sin⁡(A−30∘)sin⁡(B−30∘)sin⁡(C−30∘)>18\alpha,\beta,\gamma>30^\circ\ \Longrightarrow\ \sin(A-30^\circ)\sin(B-30^\circ)\sin(C-30^\circ)>\frac18
Detailed analysis

If alpha,beta,gamma were all greater than 30 degrees, strict decrease of each f gives the product of f_A(30), f_B(30), f_C(30) greater than 1. Multiplying by sin(30)^3 yields the displayed strict inequality.