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

Question

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 $0.5$).