## Menelaus three times

Given segment B1B2 and point A moving on curve c. On AB1, AB2 take respectively points C1, C2 such that the ratios AC1/C1B1 = r1, AC2/C2B2 = r2 are constant. Then the following is true:
1) The intersection point C of lines B1B2, C1C2 is constant.
2) Take point D on C1C2, s.t. C1D/DC2 = r3 is constant. Then the intersection point E of AD and B1B2 is constant.
3) The ratio r4 = AD/DE is also constant.

1) Apply Menelaus to triangle AB1B2 and secant CC1C2.
2) Apply Menelaus to triangle AC1C2 and secant CB1B2. It follows that CC1/CC2 = r5 is constant. Thus, the cross ratio (C,D,C1,C2) is constant. But this is the same with (C,E,B1,B2), implying that E is constant.
3) Apply Menelaus to CB2C2 and the secant ADE.
Actually the curve c (a Bezier spline) is not terribly necessary, but I leave it there since I like its shape.
Problem: Find the various ratios and cross ratios in dependence from the known ratios r1, r2 and r3.

Ceva.html
Desargues.html
Harmonic.html
LineInTrilinears.html
Menelaus.html
MenelausApp.html
MenelausFromCeva.html
Newton.html

### References

Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington DC, Math. Assoc. Ammer., 1995, p. 147.