I have an ABC triangle and I know A corner coordinates $(A_x,A_y)$ and B corner coordinates $(B_x,B_y)$. In addition I know AC side (side b) length and also BC side (side a) length of the triangel. I want to calculate C corner coordinates $(C_x,C_y)$.
I tried to use the technic about distance of two points in $2D$ space. $$ L^2=(x_1-x_2)^2+(y_1-y_2)^2$$ as follows.
Step 1: $$|AC|^2=(A_x-C_x)^2+(A_y-C_y)^2\\ |BC|^2=(B_x-C_x)^2+(B_y-C_y)^2 $$
Step 2: $$|AC|^2=A_x^2-2A_xC_x+C_x^2+A_y^2-2A_yC_y+C_y^2 \\ |BC|^2=B_x^2-2B_xC_x+C_x^2+B_y^2-2B_yC_y+C_y^2 $$
Step 3: multiply by -1 second equation $$|AC|^2=A_x^2-2A_xC_x+C_x^2+A_y^2-2A_yC_y+C_y^2\\ -|BC|^2=-B_x^2+2B_xC_x-C_x^2-B_y^2+2B_yC_y-C_y^2$$
Step 4: Sum of two equations $$|AC|^2-|BC|^2=A_x^2-B_x^2-2A_xC_x+2B_xC_x+A_y^2-B_y^2-2A_yC_y+2B_yC_y+C_y^2$$
I am stuck at this point because I can't leave alone $C_x$ or $C_y$ to substitute it at the other equation.
How can I solve this and find $C_x,C_y$ coordinates which represents 3rd coordinates of the triangle?
I have found similar questions in this forum but all of them was using the 0 (zero) value at least one or two known coordinates. For example $A_x$=0, $A_y$=5 and $B_x$=3, $B_y$=0 etc. In such a case the technic above turns into simpler to solve but in my case I am stuck. Thanks in advance?
The only thing you don't know is $C_x$ and $C_y$. Try using just $x$ and $y$ for these so you can see more easily what you are trying to solve for.