Problem 7

In the isosceles trapezoid

ABCD

with

AB\parallel DC

and

BC=AD

, let

AB=a

CD=c

, and let the perpendicular distance from

A

CD

h

. Show how to construct all points

X

on the axis of symmetry such that

\angle BXC=\angle AXD=90^\circ

. Find the distance of each such

X

from

AB

and from

CD

, and give the condition for such points to exist.

Step 5 of 5: State the construction and existence condition

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h^2\ge ac

Detailed analysis

Construct the circle with diameter $BC$ and intersect it with the symmetry axis $LM$ . There are two points when $h^2>ac$ , one tangent point when $h^2=ac$ , and no points when $h^2<ac$ . Thus the necessary and sufficient existence condition is $h^2\ge ac$ , with the distances given in Step 4.