Calculate coordinates of 3rd point (vertex) of a scalene triangle if angles and sides are known.

Solution 1:

Thanks everyone for the help! I found the answer. The formula for which is given here (Section: Intersection of two circles. It's not in the plot, but $d$ is the euclidean distance between the centers of the circles or $d=a+b$).

a = (r₀² - r₁² + d² ) / (2d)

h = r₀ sinP₀ (or) r₁ sinP₁

P₂ = P₀ + a ( P₁ - P₀ ) / d i.e., x₂ = x₀ + a (x₁ - x₀) / d (and) y₂ = y₀ + a (y₁ - y₀) / d

x₃ = x₂ ± h ( y₁ - y₀ ) / d

y₃ = y₂ ± h ( x₁ - x₀ ) / d

Solution 2:

It is enough to take the middle point of $BC$ and duplicate it. Let me explain:

Consider $M=1/2(AB+AC)$. Then $C=2BM$. Now you have the vector and its norm, so is enough to solve an equation.