Point in a triangle making a small sub-triangle

A point $P$ is picked uniformly at random inside triangle $ABC$. Joining $P$ to the three vertices creates triangles $ABP$, $BCP$ and $CAP$. What is the probability that at least one of these three has area at most a quarter of the area of $ABC$?

$\tfrac{15}{16}$. Work with the complement: every sub-triangle exceeds a quarter of the area. Each such condition keeps $P$ more than a quarter of the height from one side, and together they cut out a central triangle similar to $ABC$ with scale $\tfrac14$, so area fraction $\tfrac1{16}$.