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prove angles equal given side ratios

Source: Mongolian MO 2002 Grade 10 P6

April 9, 2021
geometryratio

Problem Statement

Let A1,B1,C1A_1,B_1,C_1 be the midpoints of the sides BC,CA,ABBC,CA,AB respectively of a triangle ABCABC. Points KK on segment C1A1C_1A_1 and LL on segment A1B1A_1B_1 are taken such that C1KKA1=BC+ACAC+ABandA1LLB1=AC+ABBC+AB.\frac{C_1K}{KA_1}=\frac{BC+AC}{AC+AB}\enspace\enspace\text{and}\enspace\enspace\frac{A_1L}{LB_1}=\frac{AC+AB}{BC+AB}.If BKBK and CLCL meet at SS, prove that ∠C1A1S=∠B1A1S\angle C_1A_1S=\angle B_1A_1S.
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