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Given a circle $o$ and distinct points $A, B,S$ outside $o$, how can I construct a circle $o'$ through $A$ and $B$, such that $S$ lies on the line joining the two intersection points of $o$ and $o'$

Any hints,

Thanks

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  • $\begingroup$ Are you sure it can be done generally? I am thinking of a counter example where o has center at (0,0), A is (1,1) and B is (1,-1). Any o' has to have center on the horizontal axis and hence intersects o, if it indeed does so, at point with identical y coordinates. That is, if S is (5,5) your o' does not exists? $\endgroup$
    – Jan
    Oct 31 '16 at 22:17
  • $\begingroup$ You need restrictions for the position of the points $A,B,S$. In many cases there are not solution, in particular when these three points are collinear. But there is a lot of many other. $\endgroup$
    – Piquito
    Oct 31 '16 at 22:20
  • $\begingroup$ It does not say generally, it just says construct a circle $\endgroup$ Oct 31 '16 at 22:21
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Well, such circle $\omicron'$ might not exist. However, there is a solution if you extend the problem in the following form: Given a circle $\omicron$ and distinct points $A, B$ and $S$ outside $\omicron$, construct a circle $\omicron'$ through $A$ and $B$, such that $S$ lies on the radical axis of the two circles $\omicron$ and $\omicron'$.

Illustration

  1. Construct the circle $k_S$ centered at $S$ and orthogonal to $\omicron$.

  2. Construct the inverse image $A'$ of $A$ with respect to $k_S$, or alternatively construct the inverse image $B'$ of $B$ with respect to $k_S$ (or you can do both it is still going to work).

  3. Draw the unique circle $\omicron'$ passing through $A, B, A'$ (or through $A, B, B'$ which is going to be the same circle $\omicron'$). In any case $\omicron'$ passes through the four points $A, A', B, B'$.

  4. If $\omicron$ and $\omicron'$ intersect at two points, then $S$ will lie on the line determined by the two intersection points of $\omicron$ and $\omicron'$. If $\omicron$ and $\omicron'$ touch at one point, then $S$ will lie on the line tangent to both $\omicron$ and $\omicron'$ at their point of contact. If $\omicron$ and $\omicron'$ do not intersect, then $S$ will lie on their radical axis.

In any case, $S$ will lie on the radical axis of $\omicron$ and $\omicron'$.

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