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Let $C_{1}$ be a circle of unit radius. Let A and B be two points inside $C_{1}$. Now I want to construct another circle $C_{2}$ such that A and B lie on $C_{2}$ and $C_{2}$ is orthogonal to $C_{1}$ at their point of intersection(I want $C_{2}$ in such a way that it intersects $C_{1}$). I tried and failed to find a way to construct such a $C_{2}$.

Any help is appreciated.

Sigur
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Phani Raj
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  • need not be..for example construct $C_{1}$ and $C_{2}$ such that they are orthogonal. Now select 2 points on $C_{2}$ whose bisector need not goes through center of $C_{1}$ – Phani Raj Jan 29 '13 at 17:16
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    My roundabout and inefficient (nay, stupid) way of doing this: apply the fractional-linear map that turns the unit disk into the upper half-plane. Now the problem is easily solved. Of course you have to transform the circle you’ve found back to the original setting. – Lubin Jan 29 '13 at 17:33
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    Take the inverse of $A$ with respect to the circle $C_1$ to obtain the third point $C$. Now construct the circle containing $A,B,C$. – Sigur Jan 29 '13 at 17:34
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    @Sigur, very elegant! – Lubin Jan 29 '13 at 17:36

2 Answers2

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Take the inverse of $A$ with respect to the circle $C_1$ (see here) to obtain the third point $C$. Now construct the circle containing $A,B,C$.

enter image description here

Community
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Sigur
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If I understand your question correctly, it is answered in Construction 2.1 of this paper: http://comp.uark.edu/~strauss/papers/hypcomp.pdf

Edit upon request: Given two points $A$ and $B$ in the interior of a circle $C_1$, invert $A$ through $C_1$ to $A'$. Now construct the circle through $A$, $B$, and $A'$. Sigur's answer contains a picture of this construction.

The paper discusses general compass-and-straightedge constructions in hyperbolic geometry, where, in particular, geodesics can be modeled as circles perpendicular to the unit circle in $\mathbb{C}$.

Neal
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