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Geometry and Trigonometry
Fig. 2.4
Fig. 2.5
Problem 2.18 Prove that in a trapezoid the midpoints of parallel sides, the point of intersection of the diagonals and the point of intersection of the non-parallel sides are collinear. Solution This is an immediate application of the above property. We have (with the notations in Fig. 2.5) [ABP ] = [CDP ],
[ABR] = [CDR],
hence R and P lie on the line through the intersection S of AB and CD found in Problem 2.17. All we have to do is to prove that [ABQ] = [CDQ]. But [ABQ] + [BQC] = [ABC] = [DBC] = [CDQ] + [BQC], and we are done. Problem 2.19 Suppose we are given a positive number k and a quadrilateral ABCD in which AB and CD are not parallel. Find the locus of points M inside ABCD for which [ABM] + [CDM] = k.