Suppose you are given a quadrilateral - orient so that one point is the "bottom" (ie. like a diamond). Given three points: the bottom point, the left point and right point, I want to solve for the top point. The only other things I know are the lengths of the top left and right sides. I know I can construct quite a few equations to describe the top point, but I'm having difficulty putting it all together.

My attempt at a solution: I can "cut" right triangles into the inside of the diamond, and from those use Pythagoras and trig. identities, but I don't know where to go from there. Eg. let the top left edge of the diamond be of length $p$, then:

$p^2 = a_p^2+b_p^2$ and $tan(\alpha)=b/a$ and $\alpha+\beta = \pi/2$ and similarly for the top right edge.

Here, $b$ is the height of the right triangle with hypotenuse $p$ and $a$ is the width; $\alpha$ the left angle and $\beta$ the top.

Where do I go from here? Thanks!

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The bottom point is irrelevant. Given two points ($P_1$ "left" and $P_2$ "right"), and the desired distances $r_1$, $r_2$ respectively of the "top" point $P_3$ from those, the "top" point must be one of the intersections of the circle with radius $r_1$ centred at $P_1$ and the circle with radius $r_2$ centred at $P_2$. You can use the law of cosines to find the angles in the triangle whose vertices are $P_1$, $P_2$ and $P_3$.