# Given a convex quadrilateral ABCD find the point K inside ABCD that divides the quadrilateral into 4 triangles with specific quality.

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 $$0.5$$).

• 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. – Bartek Jul 3 at 20:42
• 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. – Lion King Jul 5 at 6:32