Finding Triangle Vertices from Midpoints

Date: 09/18/1999 at 15:16:10
From: Wing-Shan
Subject: Midpoint coordinates

Hi, 

If you know the coordinates of the midpoints of the sides of a 
triangle, how can you find the coordinates of its vertices?

Date: 09/18/1999 at 15:50:22
From: Doctor Floor
Subject: Re: Midpoint coordinates

Hi, Wing-Shan,

Thanks for your question.

Suppose the midpoints of the sides of a triangle are given by (a1,a2), 
(b1,b2) and (c1,c2).

The centroid of the triangle formed by these midpoints is given by:

     ([a1+b1+c1]/3,[a2+b2+c2]/3)

This is also the centroid of the original triangle (can you see why?).

The centroid of a triangle divides the medians of that triangle in the 
ratio 1:2, where the side midpoints are closer to the centroid than 
the vertices. So if we make a step from (a1,a2) to 
([a1+b1+c1]/3,[a2+b2+c2]/3), and then let that step be followed by two 
steps of the same size, we find one of the vertices of the original 
triangle.

The first step from (a1,a2) to ([a1+b1+c1]/3,[a2+b2+c2]/3) is:

     [(-2a1+b1+c1)/3]
     [(-2a2+b2+c2)/3]

(this means, add (-2a1+b1+c1)/3 to the x-coordinate, and 
(-2a2+b2+c2)/3 to the y-coordinate).

The total number of steps to the vertex of the original triangle is 
three times as much:

     [ -2a1+b1+c1 ]
     [ -2a2+b2+c2 ]

We start from (a1,a2). So the vertex of the original triangle is found 
as:

     (-a1+b1+c1,-a2+b2+c2)

And in the same way the other two vertices are found as:

     (a1-b1+c1,a2-b2+c2) and (a1+b1-c1,a2+b2-c2).

I hope this helped. If have more questions, just write us back.

Best regards.
- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/