Good points in a triangle

Question

I have been trying to solve this problem with no avail . Please help

Let $ABC$ be a triangle. An interior point $P$ of $ABC$ is said to be good if we can find exactly $27$ rays emanating from $P$ intersecting the sides of the triangle $ABC$ such that the triangle is divided by these rays into $27$ smaller triangles of equal area. Determine the number of good points for a given triangle $ABC$.

Answer 1

Hint: three of the rays must pass through $A,B,C$ or you won't get $27$ small triangles. You would get $24$ triangles and $3$ quadrilaterals if none of the rays pass through a vertex of $ABC$

Answer 2

I'm not sure that the accepted answer is correct. According to this page, the answer matches the logic shown there. However, on Project Euler Problem 747, the correct answer for N=10 would be 90. However, the formula here would make it 36. Also, the answer here does not take into account the types of divisions that the diagrams show. (In particular, some of the divisions don't have all 3 vertices with rays. So based on the Project Euler data, what is the correct answer?