[alogo] 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.

[0_0] [0_1] [0_2] [0_3]

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.

See Also



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

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