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 6 of 6: Finish the angle chase
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∠ABN=∠ABC+∠NBE=32∠ABC⟹2∠AHI=3∠ABC\angle ABN=\angle ABC+\angle NBE=\frac32\angle ABC\Longrightarrow2\angle AHI=3\angle ABC
Detailed analysis

The ray BEBE lies on BCBC, so ∠ABN=∠ABC+∠NBE=32∠ABC\angle ABN=\angle ABC+\angle NBE=\frac32\angle ABC. Together with ∠AHI=∠ABN\angle AHI=\angle ABN, this gives 2∠AHI=3∠ABC2\angle AHI=3\angle ABC.