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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

the task

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

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Hint:

enter image description here

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

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  • $\begingroup$ But at (2) you can only get +2ab cos(alpha) if you use the unit circle trigonometry to my knowledge (and, as said in the original post, that's not an option). cos(180-x) = -cos(x) but that's using the unit circle (or, at least I think so). $\endgroup$ – user6115433 May 19 '16 at 19:05
  • $\begingroup$ You taught a rule of cosines for triangles with obtuse angle? $\endgroup$ – Roman83 May 19 '16 at 19:10
  • $\begingroup$ Isn't the law of cosines the same for any values of the angle? And the only reason you were allowed write +2ab cos(alpha) is that alpha and beta are supplementary (so you're actually writing -2ac cos (180-alpha) which using the unit circle transforms to +2ac cos(alpha). I have no idea how would you do that without the unit circle nor how would the angle being an obtuse help. (Also, this is 2nd-grade high school, not much special stuff is used here (and only ~2 months are dedicated to geometry)) $\endgroup$ – user6115433 May 19 '16 at 19:35

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