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Question 253208: how do you find the coordinates of a triangle's vertices by just having the coordinates of its midpoints?
Found 2 solutions by richwmiller, drk:
Answer by richwmiller(17219)   (Show Source):
You can put this solution on YOUR website! I would say you have your work cut out for yourself.
Any line connecting the midpoints is parallel to a side of the original triangle
Pick two of the points.
Find the line that goes through them.
Get the slope
Plug in an the x,y.
Use the same slope. Get the line.
Do it two more times.
Find where those lines intersect.
The intersections will be the vertices.
Answer by drk(1908)  (Show Source):
You can put this solution on YOUR website! Let A, B, and C be the verticies of the triangle.
By definition of midpoint, let
(A+B)/2 = m1
(A+C)/2 = m2
(B+C)/2 = m3,
where m1, m1, m3 are midpoints that I assume are given.
Solving using a process called Gauss-Jordan Elimination, we get
Coordinate A = m1 + m2 - m3
Coordinate B = m1 - m2 + m3
Coordinate C = -m1 + m2 + m3.
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EX: m1 = (0,2), m2 = (3,0), m3 = (3,2)
A = (0,2) + (3,0) - (3,2) = (0,0)
B = (0,2) - (3,0) + (3,2) = (0,4)
C = -(0,2) + (3,0) + (3,2) = (6,0)
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The three verticies are: (0,0) (6,0) (0,4).
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