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 5 of 6: Compute the remaining angle
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∠NBE=∠IBE=∠IBC=∠IBA=12∠ABC\angle NBE=\angle IBE=\angle IBC=\angle IBA=\frac12\angle ABC
Detailed analysis

Since IE=ENIE=EN and BE⊥INBE\perp IN, triangles IBEIBE and NBENBE are congruent. Thus ∠NBE=∠IBE=∠IBC=∠IBA=12∠ABC\angle NBE=\angle IBE=\angle IBC=\angle IBA=\frac12\angle ABC.