Associated Topics || Dr. Math Home || Search Dr. Math

### 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

- Doctor Greenie, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Triangles and Other Polygons

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search