Third Vertex of a Triangle

Question

If I have two vertices of a triangle $(x_1,y_1)$ & $(x_2,y_2)$ and I know the lengths of three sides of the triangle $a,b,c$ , how can I find the third vertex ? Is there any derived formula?

Answer 1

The third vertex will lie at the intersection point(s) of two circles centered at the two given vertices and radii equal to corresponding side lengths.

Clearly there'll be two possible positions for third vertex.

So one can write equations of two circles $S_1=0$ and $S_2=0$, equate them to get equation of common chord $L=0$ and further solve for intersection of $L$ with $S_i$'s.

Here is a diagram explaining the situation.