Theorem: Consider the midpoint F of the arc AB on the circumcircle of triangle ABC. There is a unique parabola with focus F that is tangent to all three sides of ABC; furthermore the parabola’s tangent points on sides AC and BC are two of the six points on the Conway Circle.

I’ve got an aesthetically pleasing proof using some of Archimedes’ theorems about parabolas…

]]>I did your first solution. From your observation it follows that the in-centre is at the same distance from any two consecutive points on the circle.

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