Finding Diagonals in a Quadrilateral

Question

This is just a problem I came up with while reading one of my geometry textbooks:

If we have a quadrilateral with side lengths $a, b, c$ and $ d$ and we know the length of one of the diagonals to be $x$, what is the length of other diagonal in terms of $a, b, c, d$ and $x$?

I understand that there is only one possible length for the other diagonal and could prove this using SSS congruence, but I could not figure out how to derive the length of the other diagonal based on the provided info. Please help with this.

Answer 1

Given the side lengths and the one diagonal, the law os sines gets you all the angles of the two triangles that you know. Then combine the split (by the known diagonal) angles and use the law of sines to get the length of the other diagonal. As a check, do it for both spit angles. The answers should be the same.

Law of cosines can also be used.

Answer 2

Since you know the length of one of the diagonals, as well as the lengths of all of the sides, you can use the law of cosines to compute all of the angles in the quadrilateral.

Having done this, you can use the law of cosines again to compute the length of the other diagonal.

Answer 3

For $4$ points in the plane, there is a relation between the $6$ pairwise distances between them. It says that the Cayley-Menger determinant is $0$. So let $A_i$, $i=1,4$ be $4$ points in the plane. Denote by $a=A_1A_2, b=A_2A_3$, $c=A_3A_4$, $d=A_4A_1$, $e=A_1A_3$, $f=A_2A_4$. Then we have $$ \left |\begin{matrix} 0 & a^2 & e^2 & d^2 & 1 \\ a^2 & 0 & b^2 & f^2 & 1 \\ e^2 & b^2 & 0 & c^2 & 1 \\ d^2 & f^2 & c^2 & 0 & 1 \\ 1 & 1 & 1 & 1 & 0 \end{matrix} \right| =0$$

Solving in $f^2$ gets us an equation of degree $2$. Assume now that $A_1A_2A_3A_4$ is a convex quadrilateral. Then $f^2$ will be the larger solution of the equation above. The smaller solution will be for the non-convex quadrilateral $A_1A_2A_3A'_4$, where $A'_4$ is the symmetric of $A_4$ with respect to $A_1A_3$. With some calculations we get $$ f^2 = \frac{1}{2 e^2} \left [ - e^4 + (a^2+b^2+c^2+d^2) e^2 + (a^2-b^2)(c^2-d^2) + \\+ 4 \operatorname{Area} A_1A_2 A_3 \cdot \operatorname{Area} A_1A_3 A_4 \right ]$$