Three circles
This is one of my favorite problems in Euclidean Geometry. It goes like this:
Consider three circles
that intersect in a common point
(see figure below) Circles
and
intersect also at point
Circles
intersect also at point
Circles
Consider any point
and trace a line through
and
Trace a line through
and
We want to prove that the points
and
are collinear
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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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Francisco Blanco-Silva 
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
Daniel, thanks for your proof. I just have one question: Why are those angles supplementary?
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.
A beautiful proof, Daniel. Thanks again.
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.
Neat! Would you care to contribute with a post explaining how?