Olympiad Maths Prep

Library / /3 of 29

Geometry Difficulty 4.9 AIME Prove it Ukraine

The trapezoid ABCDABCD with parallel sides BC=aBC = a and AD=2aAD = 2a is drawn on the plane. Using only ruler construct triangle with the area equals to the area of trapezoid.

Solution

We first construct the point OO of the intersection of diagonals, and the point TT of intersection of sides ABAB and CDCD. It is well known that OTOT contains midpoints PP and EE of parallel sides (fig. 11). We have AE=ED=BC=aAE = ED = BC = a. Hence ABCEABCE and BCDEBCDE are parallelograms, MM and NN are the points of intersection of their diagonals respectively. We can draw the line MNMN which is a midline of trapezoid and intersects two sides at KK and LL respectively. We then draw line BLBL, that intersects ADAD at point FF. Thus ABF\square ABF is desired, since ABF=BCL\square ABF = \square BCL.

Looking for a route rather than an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement and solution reproduced as published; topic and difficulty added by this site.