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Isosceles triangle in simple configuration

Source: Bosnia and Herzegovina TST 2006 day 2 problem 2

July 14, 2016
perpendicularisoscelesgeometry

Problem Statement

Triangle ABCABC is inscribed in circle with center OO. Let PP be a point on arc ABAB which does not contain point CC. Perpendicular from point PP on line BOBO intersects side ABAB in point SS, and side BCBC in TT. Perpendicular from point PP on line AOAO intersects side ABAB in point QQ, and side ACAC in RR. (i) Prove that triangle PQSPQS is isosceles (ii) Prove that PQQR=STPQ\frac{PQ}{QR}=\frac{ST}{PQ}
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