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Given a convex quadrilateral $ABCD$ find the point $K$ inside $ABCD$ that divides the quadrilateral into 4 triangles ($ABK$, $BCK$, $CDK$ and $DAK$). The goal is that the point $K$ is located in a way that it maximized the ratio of the radius of inscribed circle ($R_i$) over circumscribed circle ($R_o$) for all the triangles (The ratio maximum will be $0.5$).

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  • $\begingroup$ What do you mean by that? Given a side $AB$ point $K$ maximizing this ratio would have to be such that $ABK$ is equilateral and such point cannot exist for all quadrilaterals. $\endgroup$ – Bartek Jul 3 at 20:42
  • $\begingroup$ I mean the average ratio for all triangles should be maximized. The ideal case would be that all triangles are equilateral. But I think that cannot happen. So $K$ should be a point that maximize the average ratio for all 4 triangles. $\endgroup$ – Lion King Jul 5 at 6:32

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