Let ABC be any triangle. Let D be a point on side AB and let E be a point on side AC. Draw lines CD and BE and call their intersection F. Triangle ABC is then divided into three smaller triangles BDF, CEF, BCF and a quadrilateral ADFE.

Let the area of BDF equal r, the area of CEF equal q, and the area of BCF equal p. Express the area of the whole triangle ABC in terms of p, q, and r.

(In reply to easier "cheat" answer by Daniel)

As CD and BE are not given as bisectors and D and E as midpoints of AB and AC, though they could possibly be, the result is that the "cheat" method is not valid for when D and E are not the midpoints.

I, too, have yet to find a method to solve this without the inclusion of trigonometric expressions, which, given that we are to express the solution in terms of p, q, and r, should be excluded from the final solution -- as you have proposed in your first post.  I have not yet tried to confirm if it is correct, but will agree that it is ugly and hope for a more simple solution. 

Posted by Dej Mar on 2008-11-30 12:41:09