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Let $A,B,C$ lie on a straight line. $B$ is lying between $A$ and $C$. Consider all circles passing through $B$ and $C$. The point of contact of the tangents from $A$ to these circles lies on .....

We have to identify the locus not give its equation. I tried to make the diagram and apply some geometry, I found a cyclic quadrilateral with vertices : $A$, centre of circle and ends of chord of contact. But, I am unable to proceed further. Any help would be appreciated.

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The locus will be a circle.
Let the point of contact from the tangent be $P$. Note that for any circle through $B$ and $C$, $AB \times AC $ is a constant which is equal to $AP^2$(The length of the tangent squared-Tangent Secant Theorem)

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  • $\begingroup$ So elementary... Thank you. $\endgroup$
    – evil999man
    Apr 4, 2014 at 4:26

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