The following 12 lines of Maple code verify the lemma for an arbitrary triangle. The following solution is an example of conceptualizing mathematics and letting a tool like Maple implement that thinking.
There is no loss of generality if one vertex of triangle ABC is placed at the origin, and one along the positive x-axis. Point E is the interior point in triangle ABC. The areas of triangles ACE and BCE are found by the formula "half the base times height." Hence, the lengths of the altitudes from E must be found. Point E is projected onto the edges AC and BC, respectively, and the distance from E to these projections provides those heights. The ratio of the areas of triangles ACE and BCE is readily found, and is the left-hand side of the equality to be demonstrated.
Point p3, the dividing point on edge AB, is found by intersecting the line through C and E with the line through A and B. The ratio of the distances from A to p3 and p3 to B forms the right-hand side of the equality to be demonstrated.
We divide the right-hand side (the ratio of areas) by the left-hand side, in the expectation that this will be 1. And it is, modulo the appropriate simplification. I was both amazed and delighted that Maple was able to manipulate the requisite symbolic quantities without undue difficulty. I don't know if the geometry package can implement such a symbolic solution - I didn't try it because I'm much more familiar with the Lines&Planes tools in MultivariateCalculus than I am with those in the geometry package. I'd be happy to learn the extent of the power of the geometry package.
simplify(R1/R2) assuming real