Precisely two of the median lengths of a Heronian triangle are integers. The remaining median length is not an integer.

Determine the minimum possible value of the smaller of the two integer median lengths.

(In reply to Computer Solution by Justin)

From the Mathworld table given for Heronian triangles with two rational triangle medians we see that the smallest of these has sides {73, 51, 26} and rational medians {35/2, 97/2 (with the third median being 2*SQRT(949))}.

By doubling the size of this triangle there then is a triangle of sides {146, 102, 52} and rational medians, with two integer medians, {35, 97, 4*SQRT(949)}. 

The smaller of the two integer median lengths here is 35.

I have not offered proof that this is the minimum possible value, but it is less than 4550.

 

Edited on December 14, 2009, 1:32 am

Posted by Dej Mar on 2009-12-14 01:15:54