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Part of 2019 Gulf Math Olympiad

Problems(1)

gulf trapezoid geometry (triangle area, locus, angle bisector)

Source: 7th Gulf Math Olympiad 2019 GMO p1

12/31/2019
Let ABCDABCD be a trapezium (trapezoid) with ADAD parallel to BCBC and JJ be the intersection of the diagonals ACAC and BDBD. Point PP a chosen on the side BCBC such that the distance from CC to the line APAP is equal to the distance from BB to the line DPDP.
The following three questions 1, 2 and 3 are independent, so that a condition in one question does not apply in another question.
1.Suppose that Area(△AJB)=6Area( \vartriangle AJB) =6 and that Area(△BJC)=9Area(\vartriangle BJC) = 9. Determine Area(△APD)Area(\vartriangle APD). 2. Find all points QQ on the plane of the trapezium such that Area(△AQB)=Area(△DQC)Area(\vartriangle AQB) = Area(\vartriangle DQC). 3. Prove that PJPJ is the angle bisector of ∠APD\angle APD.
geometrytrapezoidLocusarea of a triangleangle bisector