# Proofs : Collinear and Distinct; Boomerang Quadrilateral

1- Prove that if AF/FB = AF'/F'B where A, B, F, F' are collinear and distinct then F does not have to equal F'

2- Suppose that the sides AB, BC, CD and DA of a quadrilateral ABCD are cut by a line at the points A' B' C' D' respectively, show that AA'/A'B * BB'/B'C * CC'/C'D * DD'/D'A = +1

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#### Solution Preview

Question 1

- prove that if AF/FB = AF'/F'B where A, B, F, F' are collinear and distinct then F does not have to equal F'

Solution to Q1:

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We need only produce one example where the conditions hold and where F is not F'

Place your points on the x-axis, <----A----F----B----F'----> and give them coordinates, say

A(0,0) F(f,0) B(b,0) F'(f',0) so 0 < f < b < f'

[I used this order after trial and error showed it would work better.]

Then AF/FB = f/(b-f) and AF'/F'B = f'/(f'-b)

Equating them to give us a condition which must ...

#### Solution Summary

Geometry proofs are provided. The solution is detailed and well presented. A diagram is included