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Given points A, B and C, Where A and C are points on circles j (with center E) and k (with center D) respectively, and the center of circle k lies on line AB, and the center of circle j lies on line CB, how can I find an equation defining center points for those circles such that line between their centers intersects them both at the same point in space?

I think there are more than one solutions, and that they are restrained to a certain domain, but I don't know how to find the equation.

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  • $\begingroup$ There are a lot of degrees of freedom here. It looks to me like you can choose $D$, $E$ and $k$’s radius freely within the given constraints and still always be able to construct the circles centered at $E$ that are tangent to $k$ (which is equivalent to the intersection condition). $\endgroup$ – amd Jan 22 '18 at 19:41
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I found a way to reformat this that makes it much easier to solve. Since the intersection point is always in line with the centers of the circles, it forms two isosceles triangles, ADK and KEC in the diagram below. From that $e + p = m$, and $2\pi = 2a + 2b + m$. Finding $a$ in terms of $b$ gives the linear equation: $a = -b + \pi - \frac{m}{2}$, where $a > BAC$ and $b > BCA$. After picking a solution from that equation, finding the point is fairly easy.

Diagram

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