MathDB
2 geometry problems, equal segments, right triangle, semicirle, angle wanted

Source: : Gulf Mathematical Olympiad 2014 p3

August 23, 2019
geometryequal segmentsright trianglesemicircleangle

Problem Statement

(i) ABCABC is a triangle with a right angle at AA, and PP is a point on the hypotenuse BCBC. The line APAP produced beyond PP meets the line through BB which is perpendicular to BCBC at UU. Prove that BU=BABU = BA if, and only if, CP=CACP = CA. (ii) AA is a point on the semicircle CBCB, and points XX and YY are on the line segment BCBC. The line AXAX, produced beyond XX, meets the line through BB which is perpendicular to BCBC at UU. Also the line AYAY, produced beyond YY, meets the line through CC which is perpendicular to BCBC at VV. Given that BY=BABY = BA and CX=CACX = CA, determine the angle ∠VAU\angle VAU.
Back to ProblemsView on AoPS