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Geometry

Source: 1994 National High School Mathematics League, Exam Two, Problem 3

March 2, 2020
geometrycircumcircleincenter

Problem Statement

Circumcircle of △ABC\triangle ABC is ⊙O\odot O, incentre of △ABC\triangle ABC is II. ∠B=60∘.∠A<∠C\angle B=60^{\circ}.\angle A<\angle C. Bisector of outer angle ∠A\angle A intersects ⊙O\odot O at EE. Prove: (a) IO=AEIO=AE. (b) The radius of ⊙O\odot O is RR, then 2R<IO+IA+IC<(1+3)R2R<IO+IA+IC<(1+\sqrt3)R.
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