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a + c < b + d, ac < bd in trapezoid with m^2 + n^2 = (a + c)^2

Source: 2003 Bosnia & Herzegovina JBMO TST p4

May 27, 2020
geometrytrapezoidgeometric inequality

Problem Statement

In the trapezoid ABCDABCD (AB∥DCAB \parallel DC) the bases have lengths aa and cc (c<ac < a), while the other sides have lengths bb and dd. The diagonals are of lengths mm and nn. It is known that m2+n2=(a+c)2m^2 + n^2 = (a + c)^2. a) Find the angle between the diagonals of the trapezoid. b) Prove that a+c<b+da + c < b + d. c) Prove that ac<bdac < bd.
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