Question 1004861
The midpoints are
(7,0),(6,5),(3,1)

{{{drawing( 600, 600, -15, 15, -10, 15,
circle(4,-4,.12),circle(10,4,.12),circle(2,6,.12),circle(3,1,.12),circle(6,5,.12),circle(7,0,.12),circle(16/3,2,.12),locate(16/3,2,M(16/3,2)),
locate(4,-4,A(4,-4)),locate(10,4,B(10,4)),locate(2,6,C(2,6)),
locate(3,1,P(3,1)),locate(6,5,Q(6,5)),locate(7,-0.3,R(7,0)),
line(4,-4,10,4),line(4,-4,2,6),line(2,6,10,4),
 graph( 600, 600, -15, 15, -10, 15, 0)) }}}

The midpoint of the triangle is {{{(2/3)s}}} the length of the median.
Find one vertex, and go {{{(2/3)s}}} the way to the midpoint of the opposite side. Directions matter. Start at the point, and take {{{(2/3)s}}} the {{{x}}} value and {{{(2/3)s}}} the {{{y}}} value.

The easiest one is {{{(2/3)s}}} from ({{{2}}},{{{6}}}) to ({{{7}}},{{{0}}})
The {{{x}}}-value is {{{(2/3)s}}} from {{{2}}} to {{{7}}}. The distance is {{{5}}}, and {{{(2/3)s}}} of that is {{{(10/3)}}}, or {{{3.33}}}. 
So the {{{x}}}-value is {{{5.33}}} or {{{16/3}}}.
 
The {{{y}}}-value is {{{(2/3)s}}} from {{{6}}} to {{{0}}}, or {{{(2/3)6=4}}}. The {{{y}}}-value is {{{2}}}.

The midpoint is ({{{16/3}}},{{{2}}}).

This will work for the other two points to the opposite side midpoint.



or, you can do it this way:

take two vertices and two midpoints and find the equations of the lines passing through one midpoint and opposite vertices

R(7,0) and  C (2,6)

*[invoke change_this_name10094 7, 0, 2, 6]

Q(6,5) and A (4,-4)

*[invoke change_this_name10094 6, 5, 4, -4]

so, your equations are:

{{{y=-1.2x + 8.4}}} and
{{{y=4.5x -22}}}
-----------------find intersection
since left sides equal, make right sides equal and solve for {{{x}}}

{{{4.5x -22=-1.2x + 8.4}}}

{{{4.5x +1.2x =22+ 8.4}}}

{{{5.7x  =30.4}}}

{{{x  =30.4/5.7}}}

{{{x  =304/57}}}

{{{x = 16/3}}}

find {{{y}}}

{{{y=4.5(16/3) -22}}}

{{{y=45(16/30) -(22*30)/30}}}

{{{y=720/30 -660/30}}}

{{{y=60/30}}}

{{{y=2}}}



intersection point is at ({{{16/3}}},{{{2}}}) and that is your midpoint


{{{drawing( 600, 600, -15, 15, -10, 15,
circle(4,-4,.12),circle(10,4,.12),circle(2,6,.12),circle(3,1,.12),circle(6,5,.12),circle(7,0,.12),circle(16/3,2,.12),locate(16/3,2,M(16/3,2)),
locate(4,-4,A(4,-4)),locate(10,4,B(10,4)),locate(2,6,C(2,6)),
locate(3,1,P(3,1)),locate(6,5,Q(6,5)),locate(7,-0.3,R(7,0)),
line(4,-4,10,4),line(4,-4,2,6),line(2,6,10,4),
 graph( 600, 600, -15, 15, -10, 15,-1.2x + 8.4,4.5x -22)) }}}