+129899 Consider one specific triangle ABC with A=(4,8), B=(2,-6), and C=(-4,4).
The midpoint of AB = [ (2 + 4) / 2 , ( 8 - 6) / 2 ] = (3, 1)
The slope of the line drawn between C and this point is [ 4 - 1] / [ -4-3] = 3/-7 = (-3/7)
So....the equation of the line joining these two points is
y =(-3/7) ( x - 3) + 1
y = (-3/7)x + 9/7 + 1
y = (-3/7)x + 16/7 (1)
Likewise....the midpoint of BC = [ (-4 + 2) / 2, (-6 + 4) / 2 ] = ( -1, -1)
And the slope of the line joining A and this point is [ 8 - - 1 ] / [ 4 - - 1 ] = 9/5
So....the line joining these two points is given by :
y = (9/5)(x - 4) + 8
y = (9/5)x - 36/5 + 40/5
y = (9/5)x + 4/5 (2)
Setting (1) and (2) equal we can find the x coordinate of the intersection point as
(-3/7)x + 16/7 = (9/5)x + 4/5
16/7 - 4/5 = (9/5 + 3/7)x
52/35 = 78/35x
52 = 78x
x = 52/78 = 2/3
And the y coordinate is (9/5)(2/3) + 4/5 = 18/15 + 4/5 = 6/5 + 4/5 = 10/5 = 2
Here is a pic :
