Dear Nikolaos, you wrote:
> If E,F,G,H are points on the sides AB, BC, CD, DA of a
> quadrilateral ABCD such that
> AE/EB = DG/GC = m
> BF/FC = AH/HD = n
> then the intersection K of EG, FH gives
> HK/KF = m
> EK/KG = n
> Does anybody knows a reference in F.G.-M. or elsewhere?
I had already seen this theorem related to skew quadrilaterals,
because AE/EB = DG/GC = m, BF/FC = AH/HD = n are sufficient
conditions for E, F, G, H to be coplanar.
I've found it explicitly stated in
"Higher Course Geometry", by H. G. Forder (n. 59, pg 76) and in
"Premier livre du tetraedre" by P. Couderc, A. Balliccioni (n. 27,
pg31), but it can be also be deduced from the proof of Theorem 663,
n. 1799, pg 875 of F. G.-M. (in my reprint of the 6th French
These all refer to skew quadrilaterals but the proofs work equally
well for the plane case.
I hope this helped