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Ratio Resolution II (Posted on 2012-11-24) Difficulty: 3 of 5
Each of x, y, z, a, b and c is a positive real number that satisfy:

(ay-bx)/c = (cx-az)/b = (bz-cy)/a

Determine with proof, the ratio x:y:z in terms of a, b and c.

See The Solution Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

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Solution I think I got it. | Comment 1 of 3
I tried assuming the numerators were all positive:
ay > bx [1a] ay/x > b [1b] a > bx/y [1c]
cx > az [2a] cx/z > a [2b] c > az/x [2c]
bz > cy [3a] bz/y > c [3b] b > cy/z [3c]
Combining [1b] with [3c] gives
ay/x > cy/z
az > cx
which is a contradiction with [3a]
the others do the same

Assuming the numerators are all negative gives the same contradictions. 

So the numerators must all be zero so the ratio x:y:z is the same as a:b:c

  Posted by Jer on 2012-11-25 00:03:57
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