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I have a triangle in the Cartesian planeˏ I know the coordinates of only two points of this triangle (none of these two points are on the origin (0,0))ˏ I also know it's sides lengths and all it's angles.

How can I find the coordinates of the third unknown point in this triangle?

Thanks a lot

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There can be two possible points, symmetrical to the side already known. – JiminP Nov 8 '13 at 16:09

Say you know the coordinates of the points $A$ and $B$. Then $C$ must lie on the circle centred at $A$ with radius $AC$. Simultaneously, it must lie on the circle centred at $B$ with radius $BC$.

You can express each of these two facts as a quadratic equation in the variables being the coordinates of $C$. If the coordinates of $A$ and $B$ and all the side lengths are compatible, this set will have exactly two solutions. Those will be the two possible locations of $C$.

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Thanks Arthur for your answer. – Mohammad Alsharif Nov 8 '13 at 16:36
Do you know how can I extract one of these two possible solutions and just adopt one? – Mohammad Alsharif Nov 8 '13 at 16:37
@MohammadAlsharif No, not really. Mathematically they both carry the same significance, and it's difficult to calculate one without more or less getting the second one for free. Traditionally you would calculate them both, then choose the one that fits you own taste best after the fact. – Arthur Nov 8 '13 at 17:19
this solution seems very unpractical. Even if I transpose the problem so that one of the endpoints is at the origin, and solve the now simple circle formula for y and plug it into the other one, the solution barely qualifies as closed form when solved algebraically, solved via wolfram alpha: – Ant6n Sep 10 '15 at 19:24

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