It is given to prove that

Every circle that passes through a fixed point and with its center on a fixed straight line, must pass through another fixed point.

As far as I have interpreted the question, what I need to prove is that

Any two circles that have a point $P$ common and have their centers lying on line $m$, they must have another common point $D$.

But as soon as I take tangential circles, I can easily disprove the very thing to be proved. Is my interpretation correct and does the question need any modification?

  • $\begingroup$ Is it possible they're referring to the other intersection point with the line as the other "fixed point"? Admittedly I'm not entirely certain of the terminology being used here $\endgroup$ Mar 25 at 10:25
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    $\begingroup$ I assume the given fixed point does not lie on the given line that the circle centres must lie on. In your supposed counterexample, how can two tangential circles centred on the given line have a point in common that is not on that line? $\endgroup$ Mar 25 at 10:39
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    $\begingroup$ I agree that the question probably intends that the given point not be on the given line. ... That said, when the point is on the line (so that all circles described are tangent), it is not-uncommon (and not-unreasonable) to count a point of tangency as a "double-point" of intersection. $\endgroup$
    – Blue
    Mar 25 at 11:51

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