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 2 of 5: Measure X along the symmetry axis

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L=\operatorname{mid}(AB),\quad M=\operatorname{mid}(CD),\quad LM=h,\quad LX=x

Detailed analysis

Let $L$ and $M$ be the midpoints of $AB$ and $CD$ . The segment $LM$ is the symmetry axis and has length $h$ . If $X$ is at distance $x$ from $L$ , then its distances from $AB$ and $CD$ are $x$ and $h-x$ , respectively.