Ok suppose you've got isoceles triangle BAC where AB=AC

Draw the medians from B and C say BX and CY (X is the midpoint of AC, Y is the midpoint of AB)

Now, there are two triangles CYB and BXC

XC=YB because they are both 1/2 the length of two equal length segments AC and AB

Because BAC is isoceles B=C

BC=CB is common to both triangles.

So you have side-angle-side of two triangles, CYB and BXC, being equal and thus the two triangles are congruent.

Thus the lengths of BX and YC are equal.

You can take all this and convert it to a 2 column proof.