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AB + AC = 3BC, midpoints and incircle related, <BIC_1 +<CIB_1 = 180^o

Source: Tuymaada Junior 2005 p7

May 12, 2019
geometryincirclemidpoints

Problem Statement

The point I I is the center of the inscribed circle of the triangle ABC ABC . The points B1 B_1 and C1 C_1 are the midpoints of the sides AC AC and AB AB , respectively. It is known that ∠BIC1+∠CIB1=180∘ \angle BIC_1 + \angle CIB_1 = 180^\circ . Prove the equality AB+AC=3BC AB + AC = 3BC
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