Given circle say A,B,C where each of them touches each other externally . We are given radius of all 3 circles. We are also given 2-D coordinates of centre of B,C ,we need to compute coordinates of A. Also given that y-coordinates of B,C are always equal to 0.
Clearly there are 2 possible coordinates for A . The direct approach writing equation using distance formula ,gives me the answer but the formula i am getting is too too complicated . If there's better solution , it will be much appreciated as i have to write a program in C++ .
A relatively easy way to get the coordinates of $A$ goes as follows:
(1) in triangle $ABC$ you know that $BC = r_B + r_C$ and similarly for $AB$ and $AC$ so you know all three sides.
(2) Use Hero's formula for the area of the triangle which is $\sqrt{s(s-a)(s-b)(s-c)}$ where $s$ is half the sum of the sides.
(3) But that same area is also $\frac12 \overline{BC}\cdot\overline{AB}\sin B$. This lets you tirivally solve for $\sin B$.
(4) The $y$ coordinate of $A$ is $\overline{AB}\sin B$.
(5) The $x$ coordinate if $B$ were the origin is $\overline{AB}\cos B$. It is easy to find that since $\cos B = \sqrt{1-\sin^2 B}$
(6) Knowing $BC$ you know where $M$ is with respect to $B$. So just shift the $x$ coordinate of everything, including $A$, by $\frac12 \overline{BC}$.