Problem
GEO-B3-M06-P024 Two Forms of One Transversal
In a complete quadrilateral, choose a triangle from three of the lines and view the fourth line as a transversal. Prove that trigonometric Menelaus for this transversal does not change if another triangle of the same complete quadrilateral is chosen.
C. Hint 1. Compare the angles made by the same line with the four sides.
D. Hint 2. Use pairwise cancellation of extra sine factors.
Let four lines form four triangles. Choose one triangle and apply trig Menelaus to the remaining line. The product contains sines of angles between the transversal and the three sides of the chosen triangle.
If a neighboring triangle is chosen, one side is replaced by another line of the complete quadrilateral, while the other two remain with opposite orientation. The ratio of the new and old products consists of pairs \(\sin\theta/\sin(180^\circ-\theta)\), each equal to \(1\). Hence the trigonometric Menelaus condition is invariant under the choice of triangle.
An abstract stability problem, useful before projective topics.