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Given

A convex quadrilateral whose sides are defined as (a=200, b=140, c=180, d=160). Diagonally (d1, d2) are equal.

Task

How to determine the length of the diagonal (d1, d2) with up to six decimal places?

Numerical results

Obtained in Mathcad using the Find function: 241.6579007199

Thanks

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    So are you satisfied with a numerical solution (because this is what you are asking for - six decimal places)? Or are you interested in how exactly your program computed this and how you can do it analytically? – 2017-02-24
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    I'm interested in an algorithm for solving the task. – 2017-02-24

1 Answers 1

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Let $\alpha=\angle(a,b)$ (the angle between the sides $a$ and $b$) and $\beta=\angle(a,d)$. For any given angle $\alpha$ you can compute exactly one angle $\beta$ (using basic geometry), so that the side $c$ has the desirded length and the quadrilateral is convex. So, essentially, you have a single parameter to determin: $\alpha$. Now use your favorite optimizations scheme to find $\alpha$ in such a way, that the diagonals are of equal length, e.g.:

These procedures are searching for a zero of a function in a single variable. So for example try to find the zero of $f(\alpha):=|d_1(\alpha)-d_2(\alpha)|$, which is the difference in the length of the diagonals. If this functions is zero, the diagonals are of the same length.

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    In this quadrangle all angles of vertices differ (see. [figure](https://drive.google.com/open?id=0B2Gy5OzgV1IbbFpHZ1dCMkk4OUU)). – 2017-02-24
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    I applied a decision by the ratio of the square (see. [example](https://drive.google.com/open?id=0B2Gy5OzgV1IbWUR3ZmxVSjA0S3c)) – 2017-02-24
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    This task has a purely algebraic solution (see. [cubic equation](https://drive.google.com/open?id=0B2Gy5OzgV1IbT3c5Yi1UTVpGMnM)) – 2018-03-06