If the sides of a Tetragon pass through the vertices of a Parallelogram, and if three of them are bisected at those vertices: prove that the fourth is so also.
Let ABCD be the Tetragon; and let the 3 sides, AB, BC, CD, be bisected by vertices of the Parallelogram EFGH.
Join BD.
∵ in Triangle BCD, sides BC, CD are bisected at F and G,
∴ FG is parallel to BD;
but EH is parallel to FG;
∴ EH is parallel to BD;
∴ Triangles AEH, ABD are similar;
now AE is half of AB;
∴ AH is half of AD.
Q.E.D.