ABCD ia parallelogram and APQ is a straight line meeting BC at P and DC produced at Q. Prove that the rectangle obtained by BP and DQis equal to the rectangle contained by AB and BC.
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The diagram shows parallelogram ABCD and the line APQ.
ΔABP and ΔCQP are similar, as the corresponding sides are parallel. Hence, their ratios:
CQ / AB = PQ / AP = CP / BP
As AB = CD, Adding 1 to each of the above ratios, we get
(CD+CQ) / AB = (PQ + AP) / AP = (CP + BP) / BP
DQ / AB = AQ / AP = BC / BP
=> BP * DQ = AB * BC
ΔABP and ΔCQP are similar, as the corresponding sides are parallel. Hence, their ratios:
CQ / AB = PQ / AP = CP / BP
As AB = CD, Adding 1 to each of the above ratios, we get
(CD+CQ) / AB = (PQ + AP) / AP = (CP + BP) / BP
DQ / AB = AQ / AP = BC / BP
=> BP * DQ = AB * BC
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