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I know length of a convex quadrilateral's four sides, and one diagonal. Then, how to calculate the other diagonal's length? Here's an example.(Looking for BD)

Convex Quadrilateral

Convex Quadrilateral

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Using the theorem of cosines we obtain $$BD^2=a^2+d^2-2ad\cos(\alpha)$$

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If you know the four sides and a diagonal (e.g.$AC$) then you know the area by applying the Heron's formula to the two triangles.
Then applying the same formula to the two triangles with the unknown diagonal in common, you can solve for that.

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