Pairs of Segments with

Equal Sums

Given AC, CB, AD, DB such that AC + CB = AD + DB.

Extend AC and DG to intersect at point X.

Extend AD and CG to intersect at point Y.


Prove that AX + XB = AY + YB.

Comment --

If we consider points A and B as the foci of an ellipse, then by the definition of an ellipse, that it is the locus of points with distances from two foci a constant sum, then points C and D are points on the ellipse.

Does this help? Is there any argument that points X and Y will be points on an ellipse with focal points A and B?

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