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Merry Christmas, Euclid...

Though this problem probably can be solved in multiple ways, one solution approach helps explain its title.

The Given:
Triangle ABC
Ray AX bisects angle BAC, intersecting side BC at X

Prove:
BX/XC = AB/AC

 

Source: Math Gazette, December 1974


Hint: Construct AB'C', the mirror image of triangle ABC in AX. Then...

Note: With the diagram arranged with point A at the top of your paper and by constructing segment CC', either a "xmas" tree or "star" shape is created by the symmetry.

 


Solution Commentary: The solution involves a good use of proportional relationships.

Construct segment BB' and CC'
By similar triangles BB'X and CC'X, BX/XC = BB'/C'C
By similar triangles ABB' and AC'C, BB'/C'C = AB/AC'
But, AC' = AC implies by transitivity that BX/XC = AB/AC