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Square with Area Equal to a Triangle

```
Date: 10/23/1999 at 11:53:34
From: Christina
Subject: Areas of triangles and squares

Given an arbitrary triangle, how can you construct a square with the
same area as the triangle? I just can't seem to figure it out.
```

```
Date: 10/27/1999 at 05:06:46
From: Doctor Floor
Subject: Re: Areas of triangles and squares

Hi Christina,

Given any triangle, construct altitude ha perpendicular to side a.
Then bisect side a.

We now have two segments of lengths a/2 and ha respectively. The
product of these lengths is equal to the area of the given triangle.

Now create the following situation:

D
|                AB = ha     BC = a/2
|                D is on a circle with AC as diameter
|                BD is perpendicular to AC
A-----B-----------C

We have BD/AB = BC/BD = tan(<DAC), and thus BD is the geometric mean
of AB and BC. Or, stated differently, BD = sqrt(AB*BC).

And thus BD is the sidelength of the square you are looking for.

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 Geometry
High School Triangles and Other Polygons

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