# Contruct a rectangle with ruler compass with one edge of a given length and area equal to a given triangle

This is a question in Geometry by Hartshorne Exercise 3.3

The goal is using Ruler and compass and a given triangle ABC and given a segment DE, construct a rectangle with content equal to the triangle ABC, and with one side equal to DE. Any propositions in Euclid book I-IV are usable but its likely going to be in book II as thats where all the area postulates are i got the given rectangle construct a square of the same area but can't get this one.

I realize there is the exact question with answers but they use non-ruler and compass methods (thought i believe that is what OP actually wanted but didn't specify)

This is a special case of Euclid I, 44: "To a given straight line to apply, in a given rectilineal angle, a parallelogram equal to a given triangle." The given angle here is a right angle, and the parallelogram is a rectangle.

Given the base $a$ and height $h$ of the triangle, and the length $x$, you want to construct length $y$ such that $a\frac {h}{2}=xy$. Since this is equivalent to $x:\frac {h}{2}=a:y$, draw any angle $\angle POQ$ where $OP=x$, $OQ=\frac {h}{2}$, draw a point $R$ on the line $OP$ such that $OR=a$, and finally draw a parallel to $PQ$ through $R$ so that it intersects the line $OQ$ in $S$. You will get, by similarity, $OS=y$.