Given segment B_{1}B_{2} and point A moving on curve c. On AB_{1}, AB_{2} take respectively points C_{1}, C_{2} such that the ratios AC_{1}/C_{1}B_{1} = r_{1}, AC_{2}/C_{2}B_{2} = r_{2} are constant. Then the following is true:

1) The intersection point C of lines B_{1}B_{2}, C_{1}C_{2} is constant.

2) Take point D on C_{1}C_{2}, s.t. C_{1}D/DC_{2} = r_{3} is constant. Then the intersection point E of AD and B_{1}B_{2} is constant.

3) The ratio r_{4} = AD/DE is also constant.

1) Apply Menelaus to triangle AB_{1}B_{2} and secant CC_{1}C_{2}.

2) Apply Menelaus to triangle AC_{1}C_{2} and secant CB_{1}B_{2}. It follows that CC_{1}/CC_{2} = r_{5} is constant. Thus, the cross ratio (C,D,C_{1},C_{2}) is constant. But this is the same with (C,E,B_{1},B_{2}), implying that E is constant.

3) Apply Menelaus to CB_{2}C_{2} 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 r_{1}, r_{2} and r_{3}.

### See Also

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.

Return to Gallery