Pillow-Problems: Problem #3

Problem #2 | Problem #4
For more information on this collection, see Pillow-Problems by Charles L. Dodgson (Lewis Carroll).

Problem:

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.

Solution:

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.