MathLabs

Problem 1

Let ABC ABC be an acute-angled triangle with circumcentre O O . Let P P on BC BC be the foot of the altitude from A A . Suppose that ∠BCA≥∠ABC+30∘\angle BCA\ge\angle ABC+30^\circ . Prove that ∠CAB+∠COP<90∘\angle CAB+\angle COP<90^\circ .
Step 2 of 4: Compare midpoints
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In plain words

A large central angle gives a long half-chord.

AZ=YP≥R/2AZ=YP\ge R/2
Detailed analysis

Let Z,Y Z,Y be the midpoints of AD,BC AD,BC . Since AD≥R AD\ge R , AZ≥R/2 AZ\ge R/2; the parallel chords form a rectangle, so AZ=YP AZ=YP .