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many angleswith point on the arc of circumcircles

Source: JBMO Shortlist 2017 G3

July 25, 2018
geometrycircumcircleequal anglesangles

Problem Statement

Consider triangle ABCABC such that AB≤ACAB \le AC. Point DD on the arc BCBC of thecircumcirle of ABCABC not containing point AA and point EE on side BCBC are such that ∠BAD=∠CAE<12∠BAC\angle BAD = \angle CAE < \frac12 \angle BAC . Let SS be the midpoint of segment ADAD. If ∠ADE=∠ABC−∠ACB\angle ADE = \angle ABC - \angle ACB prove that ∠BSC=2∠BAC\angle BSC = 2 \angle BAC .
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