# parallelogram diagonals in a relationship with basic geometry

This was a question in my textbook for homework a while ago but not even the teacher can find the solution using only basic geometry (further rules below). Basic only since it's in the section where we don't know about vectors or the unit circle (it's easy with the unit circle) at that point.

## Rules

• Allowed trigonometric functions in a right triangle (and any transformations of them based on the right triangle, however, no unit circle)
• Law of sines and cosines allowed, along with heron's formula
• Basic information allowed (diagonals split in each other in $2$, sum of angles, what is parallel and what not and such)
• No vectors (or rules that come from them such as the parallelogram rule) or unit circle

You are given a parallelogram $ABCD$. $|AB|$ is equal to $23$ units, $|BC|$ is equal to $11$ units. The diagonals are in a $3/2$ relation; $f/e = 2/3$. Find alpha (angle with diagonal $e$ out of it; any angle is fine though) and the length of both diagonals.

Sketch of the parallelogram

Imgur mirror:http://i.imgur.com/fhOhoOV.png

Let $f=2x, e=3x$. $$2(a^2+b^2)=e^2+f^2$$ $$2(23^2+11^2)=4x^2+9x^2$$