Triangle Midpoints and Vertices

Date: 02/23/2003 at 11:54:32
From: Katie
Subject: Geometry

How do you figure out the vertices of a triangle algebraically by 
using its three midpoints?

Date: 02/23/2003 at 20:43:30
From: Doctor Greenie
Subject: Re: Geometry

Hi, Katie -

This problem is solved very neatly using this fact:

  In any triangle, the line joining the midpoints of two sides is
  parallel to the third side and half the length of that third side.

Then, given the midpoints A, B, and C of the three sides of a 
triangle, we know that

(1) the two endpoints of the side of the triangle with midpoint A lie 
on a line containing A that is parallel to BC; the distance of each 
endpoint from point A is equal to the length of BC.

(2) the two endpoints of the side of the triangle with midpoint B lie 
on a line containing B that is parallel to AC; the distance of each 
endpoint from point B is equal to the length of AC.

(3) the two endpoints of the side of the triangle with midpoint C lie 
on a line containing C that is parallel to AB; the distance of each 
endpoint from point C is equal to the length of AB.

A specific example shows that this method, while its description 
might sound somewhat complicated, is in fact very easy to use.  
Suppose the three given midpoints are A(-1,2), B(5,5), and C(3,-2).

Let's find the endpoints of the side of the triangle with A as its 
midpoint, using (1) above. We need P and Q such that PA and QA are 
each parallel to BC and the same length as BC. Speaking informally, 
to get from B to C we have to move left 2 and down 7; so one of the 
endpoints of the side of the triangle with A as its midpoint is left 
2 and down 7 from A(-1,2) - i.e., at (-3,-5). And to get from C to B 
we have to move right 2 and up 7; so the other endpoint of the side 
of the triangle with A as its midpoint is right 2 and up 7 from 
A(-1,2) - i.e., at (1,9).

To find the endpoints of the side of the triangle with B as its 
midpoint, we see that to get from A to C we move right 4 and down 4; 
so to find the endpoints of the side of the triangle with B as its 
midpoint, we move right 4 and down 4 from B to get to (9,1), and we 
move left 4 and up 4 from B to get to (1,9).

And to find the endpoints of the side of the triangle with C as its 
midpoint, we see that to get from A to B we move right 6 and up 3; so 
to find the endpoints of the side of the triangle with C as its 
midpoint, we move right 6 and up 3 from C to get to (9,1), and we 
move left 6 and down 3 from C to get to (-3,-5).

So the three vertices of the original triangle, with midpoints 
A(-1,2), B(5,5), and C(3,-2), are (-3,-5), (1,9), and (9,1). Notice 
that we determined each of these vertices twice in our process; this 
gives us the opportunity to catch any arithmetic errors we might 
make. (The two answers we get for each vertex must be identical; if 
they are not, we made a mistake in our arithmetic.)

For two algebraic approaches to this same problem (which are not only 
very different from the answer above but also very different from 
each other), take a look at the following pages in the Dr. Math 
archives:

   Finding Triangle Vertices from Midpoints
   http://mathforum.org/library/drmath/view/55193.html 

   Using Midpoints to Determine Vertices
   http://mathforum.org/library/drmath/view/61165.html 

It is curious to note that one of these links shows an answer I 
myself provided to a similar question some time ago.  Looking back at 
that answer (which uses an algebraic approach), it seems to me that 
the visual/geometric approach in the answer above is much easier to 
use.  But you (or any other reader) might prefer an algebraic 
approach....

I hope all this helps.  Please write back if you have any further 
questions about any of this.

- Doctor Greenie, The Math Forum
  http://mathforum.org/dr.math/