MathLabs

Problem 2

Let ABCABC be an acute triangle with ∠BAC=60∘\angle BAC=60^\circ and AB>ACAB>AC. Let II be its incenter and HH its orthocenter. Prove that 2∠AHI=3∠ABC2\angle AHI=3\angle ABC.
Step 4 of 6: Convert the target angle
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∠AHI=180∘−∠IHK=180∘−∠AKN=∠ABN\angle AHI=180^\circ-\angle IHK=180^\circ-\angle AKN=\angle ABN
Detailed analysis

From the isosceles trapezoid and the cyclic quadrilateral, ∠AHI=180∘−∠IHK=180∘−∠AKN=∠ABN\angle AHI=180^\circ-\angle IHK=180^\circ-\angle AKN=\angle ABN.