[alogo] Thales theorem application

Given trapezium ABCD extend the parallel side DC by equal segments HD = CI and draw lines EH, EI, E being the intersection point of non-parallel sides AD and BC. Then intersect lines EH, EI with the diagonals BD and AC respectively, to obtain segment FG. Show it to be parallel to AB.

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HIBA is a trapezium having middles of parallel sides the same with those of trapezium ABCD. Hence the line of middles of two trapezia is the same line JK. By Thales theorem (see Thales2.html ) implies that F'G' is parallel to AB.
Then work with the two trapezia HDG1G', CIF'F1 having respectively equal parallel sides HD=CI and G'G1=F1F', to show that GF is parallel to HI.

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