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 1 of 4: Write the perpendicular-distance identities
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PX=BXsin⁡β=CXsin⁡(C−γ),QX=CXsin⁡γ=AXsin⁡(A−α),RX=AXsin⁡α=BXsin⁡(B−β)PX=BX\sin\beta=CX\sin(C-\gamma),\quad QX=CX\sin\gamma=AX\sin(A-\alpha),\quad RX=AX\sin\alpha=BX\sin(B-\beta)
Detailed analysis

Let alpha=XAB, beta=XBC, gamma=XCA, and let P,Q,R be the feet from X to BC,CA,AB. Right triangles at the relevant sides give the three displayed equalities, where A,B,C denote the angles of triangle ABC.