The sides of a trapezoid are 5, 8, 11, and 13, and its diagonals are also integer numbers; what are they?

I also wrote a program testing all cases using the law of cosines.  I similarly found no match.

Here's a summary of my program:

-Let a,b,c,d be the sides of the trapezoid, with a and c being the parallel sides.  For each possible assignment of these variables:
     -Compute the quantity q=ab^2+ac^2+cb^2+ca^2.
     -Let x and y independently range from 1 to floor(sqrt(q)).  These will represent the diagonals.  For each assignment of x and y:
          -Compute the quanity p=ax^2+cy^2.
          -If p equals q, print as a candidate solution.

No candidate solutions were printed. 

The equation p=q must follow from the following derivation:

x^2 = b^2 + c^2 - 2bc*cos(theta)
y^2 = b^2 + a^2 +2ab*cos(theta)

(using the fact that the angles surrounding side b are theta and 180-theta)

Multiply the first equation by a, the second by c, and add, to get:

p=q.

Posted by David Shin on 2005-01-17 18:04:04