Trapezoid midpoints, midlines and harmonic mean

Question

Trapezoid ABCD has parallel sides AB and CD, of lengths a and b respectively. Diag-

onals AC and BD intersect at E. Draw the line through E that is parallel to AB and CD,

and let P and Q be its intersections with AD and BC respectively.

(a) Prove that E is the midpoint of PQ.

(b) Show that PQ = 1/ [1/2(1/a+1/b)]. PQ is known as the harmonic mean of a and b.

You have seen that the midline of a trapezoid does not divide the trapezoid into two

similar trapezoids. Is it possible that a different line (parallel to the midline) could divide

the trapezoid into two similar trapezoids?

Answer 1

(a) Use diagonals of a trapezoid are mutually bisecting. Then congruent triangles.

I question the accuracy of the statement:

You have seen that the midline of a trapezoid does not divide the trapezoid into two

similar trapezoids.

Three similar trapezoids are formed by AAAA.