Question 1197771
<br>
Let the given points be A(0,-3), B(-0.5,3.5), and C(-3.5,0.5).<br>
Draw the triangle with those points as vertices.<br>
{{{drawing(400,400,-5,5,-8,8,
line(0,-3,-0.5,3.5),line(-0.5,3.5,-3.5,0.5),line(-3.5,0.5,0,-3)
,locate(0.5,-3,"A(0,-3)"),locate(0,3.5,"B(-0.5,3.5)"),locate(-3,0.5,"C(-3.5,0.5)")
)}}}
The lines joining the midpoints of a triangle are parallel to and half the length of the sides of the triangle.  So we can find the vertices of the triangle we want by copying any of the sides of triangle ABC in both directions from the opposite vertex.<br>
For example, segment BC can be described informally as "left 3, down 3".  We can find two of the vertices of the triangle we are looking for by going "left 3, down 3" from A and by going "right 3, up 3" from vertex A:<br>
{{{drawing(400,400,-5,5,-8,8,
line(0,-3,-0.5,3.5),line(-0.5,3.5,-3.5,0.5),line(-3.5,0.5,0,-3)
,locate(0.5,-3,"A(0,-3)"),locate(0,3.5,"B(-0.5,3.5)"),locate(-3,0.5,"C(-3.5,0.5)")
,green(line(0,-3,-3,-6)),green(line(0,-3,3,0))
,locate(-2.5,-6,"D(-3,-6)"),locate(3,0,"E(3,0)")
)}}}<br>
Then, to find the third vertex of the triangle we are looking for, we can either copy segment AC starting from B, or copy segment AB starting from C.  The next drawing shows segment AC ("left 3.5, up 3.5") copied starting from B:<br>
{{{drawing(400,400,-5,5,-8,8,
line(0,-3,-0.5,3.5),line(-0.5,3.5,-3.5,0.5),line(-3.5,0.5,0,-3)
,locate(0.5,-3,"A(0,-3)"),locate(0,3.5,"B(-0.5,3.5)"),locate(-3,0.5,"C(-3.5,0.5)")
,green(line(0,-3,-3,-6)),green(line(0,-3,3,0))
,locate(-2.5,-6,"D(-3,-6)"),locate(3,0,"E(3,0)")
,green(line(-0.5,3.5,-4,7)),locate(-4,8,"F(-4,7)")
)}}}<br>
We have found the three vertices of the triangle we are looking for; we can show the triangle by drawing the missing segments.<br>
{{{drawing(400,400,-5,5,-8,8,
line(0,-3,-0.5,3.5),line(-0.5,3.5,-3.5,0.5),line(-3.5,0.5,0,-3)
,locate(0.5,-3,"A(0,-3)"),locate(0,3.5,"B(-0.5,3.5)"),locate(-3,0.5,"C(-3.5,0.5)")
,green(line(0,-3,-3,-6)),green(line(0,-3,3,0))
,locate(-2.5,-6,"D(-3,-6)"),locate(3,0,"E(3,0)")
,green(line(-0.5,3.5,-4,7)),locate(-4,8,"F(-4,7)")
,green(line(-3,-6,-3.5,0.5)),green(line(-3.5,0.5,-4,7)),green(line(3,0,-0.5,3.5))
)}}}<br>