Hello, everyone!
This is the long and clunky solution I posted elsewhere . . .
Construct a convex quadrilateral and then a quadrilateral similar to it
whose area is three-fourths of the area of the original quadrilateral.
Suppose the original quadrilateral has sides $\displaystyle a,b,c,d.$
The smaller quadratilateral has sides multiplied by a factor of $\displaystyle \frac{\sqrt{3}}{2}.$
Construct an equilateral triangle with side 1.
Contruct an altitude; its length is $\displaystyle \frac{\sqrt{3}}{2}$
(I'll leave the proof to you.)
On a line, mark off points $\displaystyle A,\,B,\,C,\,\text{ where }AB \,=\,1,\:BC \,=\,\frac{\sqrt{3}}{2}$
From $\displaystyle \,A$, draw a line to the upper-right.
Code:
*
*
*
*
*
*
*
*
*
*
*
o-----o----o
A B C
On that line, mark off $\displaystyle AP \,=\,a$
Code:
*
*
*
*
P *
o
*
a *
*
*
*
o-----o----o
A B C
Draw line segment $\displaystyle BP.$
Code:
*
*
*
*
P *
o
*/
a * /
* /
* /
* /
o-----o----o
A B C
From $\displaystyle \,C$ construct $\displaystyle CQ\,\parallel\, BP$
Code:
o Q
*/
* /
* /
P * /
o /
a */ /
* / /
* / /
* / /
* / /
o-----o----o
A B C
Then: $\displaystyle PQ \:=\:\frac{\sqrt{3}}{2}\,a \:=\:a'$
Repeat the process with side $\displaystyle \,b$ and find $\displaystyle b'.$
With that information, you can complete the construction.
Code:
a
* * * * * * *
* a' * *
* * *
* b' d' * * d
b * * *
* c' * *
* * * * * * * * * * * * * *
* *
* * * * * * * * * * * * * * * * * * *
c
Note that: .$\displaystyle c'\parallel c\,\text{ and }\,d'\parallel d.$