Contribution for Puzzlet #062

From:  Denis Borris [borrisd@ca.inter.net]

AB = x+1, AC = x-1, BC = x, AD = x-2:

        A

B (x-a) D (a) C

(not to scale! worse than yours)

Let DC = a; then BD = x-a;
from the 2 right triangles ABD and ACD:
(x+1)^2 - (x-a)^2 = (x-1)^2 - a^2
solve to get: a = (x-4) / 2

Using triangle ADC and substituting above:
(x-2)^2 + [(x-4)/2]^2 = (x-1)^2
solve to get: x^2 - 16x + 28 = 0
factor that sumbitch: (x - 14)(x - 2) = 0
So: x = 14 and x = 2

So answer is 14: 2 rights, 9-12-15 and 5-12-13,
forming a 13-"14"-15 triangle of height 12.

By the way, 2 is also valid solution, since you did not
specify that area  0

Well, I'll be darned! I just didn't see that one coming! Very neat analysis.  I'm almost tempted to withdraw the Puzzlet as the idea is to set problems that need a computer program to solve them.  However, I think a lot of readers will still enjoying solving it the hard way. Thanks for the excellent input - I'll add it to the contributions page.

Dave.

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