Three circles

This is one of my favorite problems in Euclidean Geometry. It goes like this:

Consider three circles c_1, c_2, c_3 that intersect in a common point O. (see figure below) Circles c_1 and c_2 intersect also at point A. Circles c_2 and c_3 intersect also at point B. Circles c_1 and c_3 intersect also at a point C. Consider any point P \in c_1, and trace a line through P and A. This line intersects circle c_2 at a second point P'. Trace a line through P' and B. This line intersects circle c_3 at a second point P''. We want to prove that the points P, P'' and C are collinear

Miscellaneous

See the post Using tikz as an IGSE for an explanation on how to produce interactive geometric diagrams ala Cabri or Geometer's Sketchpad

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  1. Daniel Maxin
    December 12, 2010 at 9:00 am

    OCP is the suplementary angle of OAP hence OCP=OAP’
    OCP” is the suplementary agle of OBP” hence OCP”=OBP’
    But OAP’+OBP’=180 degrees hence OCP+OCP”=180

    • December 13, 2010 at 5:40 pm

      Daniel, thanks for your proof. I just have one question: Why are those angles supplementary?

      \begin{array}{c}\angle OCP + \angle OAP = \pi?\\ \angle OCP'' + \angle OBP'' = \pi? \end{array}

      • Daniel Maxin
        December 13, 2010 at 8:39 pm

        This is known to happen in a quadrilateral inscribed in a circle, i.e. the sum of the opposite angles is 180. This is because an angle inscribed in a circle has a measure equal to half the radian measure of the subtended arc. When we add the opposite angles mentioned above we obtain the full circle, half of which is 180 degrees.

        • December 13, 2010 at 8:52 pm

          A beautiful proof, Daniel. Thanks again.

  2. Alison
    December 12, 2010 at 9:20 am

    If instead you find the point O, then you’re doing exactly what’s done to find Earthquakes. In fact it’s one of my favourite outreach exercises.

    • December 12, 2010 at 9:32 am

      Neat! Would you care to contribute with a post explaining how?

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