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### Deriving the Midpoint Formula

```
Date: 08/08/99 at 16:04:07
From: Jason
Subject: Midpoint formula (proof)

I'm wondering how the midpoint formula is derived from the distance
formula. Is there a better way to prove the midpoint formula?

Sincerely,
Jason
```

```
Date: 08/09/99 at 04:08:48
From: Doctor Floor
Subject: Re: Midpoint formula (proof)

Hi Jason,

Let us consider the points P1 = (x1,y1) and P2 = (x2,y2). For ease of
use, we can rewrite P2 = (x2,y2) = (x1+2t,y1+2u). This second way of
writing is easy, because we immediately see that, following the
"midpoint formula," the midpoint should be M =(x1+t,y1+u). We can also
immediately see that the slope of the line P1P2 is equal to the slope
of the line P1M, so the three points are on one line.

The distance formula gives:

d(P1,P2) = sqrt((2t)^2 + (2u)^2)
= sqrt( 4(t^2 + u^2) )
= 2 sqrt(t^2 + u^2)

while

d(P1,M) = sqrt(t^2 + u^2)

so, indeed the distance d(P1,M) = d(P1,P2)/2. As desired.

But I think it is much easier to use geometry for the proof. We can
use the fact that if we perpendicularly project a segment onto an
image segment on a given line, then the midpoint of the original
projects to the midpoint of the image. When we apply this on segment
P1P2, and perpendicularly project it to the x-axis and y-axis
respectively, we immediately find the midpoint formula. See the
following figure:

P2
/|
/ |
/  |
/   |
/    |
M     |
/|     |
/ |     |
/  |     |
/   |     |
/    |     |
P1     |     |
|     |     |
|     |     |
---------x1-----*-----x2--------- x-axis
|
(x1+x2)/2

If you need more help, just write us back.

Best regards,
- Doctor Floor, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Euclidean/Plane Geometry
High School Geometry

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