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Help with proving an identity.

I recently found this problem in my maths book, and I don't know how I should solve it:

Attachment 20790 *Show that $\displaystyle L^2=(ab+cd)(ac+bd)/(bc+ad)$*

I looked in the answers section of the book, and there it said that I should begin by using the law of cosines on the triangles ABD and BDC, and then use the fact that $\displaystyle \cos C= -\cos A$, as $\displaystyle A+C=180$ degrees. Why is that? Have I forgotten about some basic geometric rule or something?

And still, even after using the fact above, I couldn't complete the task. So how do I complete this task?

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