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 2 of 6: Use the kite symmetry
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BICN is a kite,IE=ENBICN\text{ is a kite},\qquad IE=EN
Detailed analysis

The equal angles and IN⊥BCIN\perp BC show that BICNBICN is a kite with symmetry axis ININ. Therefore EE is the midpoint of BCBC and IE=ENIE=EN.