Problem

Source: Chinese TST 2009 1st quiz P1

Tags: geometry, circumcircle, trigonometry, geometric transformation, homothety, reflection, incenter



Given that circle $ \omega$ is tangent internally to circle $ \Gamma$ at $ S.$ $ \omega$ touches the chord $ AB$ of $ \Gamma$ at $ T$. Let $ O$ be the center of $ \omega.$ Point $ P$ lies on the line $ AO.$ Show that $ PB\perp AB$ if and only if $ PS\perp TS.$