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equal angles in an isosceles triangle, line passes thought midpoint of base

Source: SRMC 2011

September 2, 2018
geometryequal anglesisoscelesmidpoint

Problem Statement

Given an isosceles triangle ABCABC with base ABAB. Point KK is taken on the extension of the side ACAC (beyond the point CC ) so that ∠KBC=∠ABC\angle KBC = \angle ABC. Denote SS the intersection point of angle - bisectors of ∠BKC\angle BKC and ∠ACB\angle ACB. Lines ABAB and KSKS intersect at point LL, lines BSBS and CLCL intersect at point MM . Prove that line KMKM passes through the midpoint of the segment BCBC.
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