# Finding three collinear points passes through three circles

Assume that we have three collinear points
$A(x_0,y_0),B(x_1,y_1)$ and $C(x_2,y_2)$.

They are on three different circles whose centres and radii are respectively
$\big((P_x, P_y), r_P\big)$, $\big((Q_x, Q_y), r_Q\big)$ and $\big((R_x, R_y), r_R\big)$

Name The euclidian pairwise distances between three points are known.
How do I calculate if there is a unique solution for $(x_i, y_i)$ pairs s.t. $i = 0,1,2$?

• These are circles in space, or spheres? – Alan May 13 '14 at 19:43
• Sorry for the wrong notation, they are all in 2-D. – padawan May 13 '14 at 20:19
• So you want to know if a line intersects each circle at exactly one point? Is that correct? – AnonSubmitter85 May 13 '14 at 20:36
• @AnonSubmitter85 this is true. And I know the distances of those points. – padawan May 13 '14 at 20:59
• Well, if the line intersects the circle at one and only one point, then the orthogonal distance from the center of the circle to the line must equal the radius of the circle. If this distance is greater than the radius, then there is not intersection; it's less than the radius, then there are two points where line and circle meet. You don't need the distance between the collinear points. At least I don't think you do. – AnonSubmitter85 May 13 '14 at 21:08