We know that, there exist integer triangles with three rational medians. The smallest has sides ($68, 85, 87$). Others include ($127, 131, 158$), ($113, 243, 290$), ($145, 207, 328$) and ($327, 386, 409$). For example, How to find coordinates of vertices of an integer triangle has sides ($68, 85, 87$)? Please help me to get them. I don't know how to start?
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I don't see what rational medians has to do with it. If there is a triangle with sides $a,b,c$ (integer or not), you can put one vertex at $(0,0)$, a second one at $(a,0)$, and then the third at $(x,y)$, and solve the equations $x^2+y^2=b^2$, $(x-a)^2+y^2=c^2$.
You can just place two vertices, say one at $(0,0)$ and one at $(0,68)$. Then if the other one is at $(x,y)$ we know $x^2+y^2=85^2, (x-68)^2+y^2=87^2$ You can expand the square and subtract the equations to get $-2\cdot 68 x+68^2=87^2-85^2$, which is a linear equation.