Let an equilateral triangle have the length of each side an integer $N$. I need to find if it is possible to transform the triangle keeping two sides fixed and alter the third side such that it still remains a triangle, but the altered side will have its length as an even integer, and the line drawn from the opposite vertex to the mid-point of the altered side is of integral length, i.e. it becomes an isosceles triangle.
Example : If $N=5$ then the answer is YES while if $N=3$ answer is NO.
It's a computer graphics problem that is a sub-part of bigger problem, I have been racking brains about maths and the concept behind it, to solve my problem.
To restate your question more clearly, you are asking what kind of integer number $N$ can be the hypothenuse of a right-angled triangle, so that the other two sides are of integer length too.
The answer is well known: $N$ must be (the multiple of) the sum of two perfect squares: $N=a^2+b^2$ (or $N=k(a^2+b^2)$). If so, the other two sides (which in your problem are half the modified side and "the line drawn from the opposite vertex to the mid-point of the altered side") are given by $a^2-b^2$ and $2ab$ (multiplied by $k$ if needed).
Example: $5=2^2+1^2$, $3=2^2-1^2$, $4=2\cdot(2\cdot1)$.