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### Triangle Construction Given Two Angles and Semiperimeter

```
Date: 03/14/2002 at 22:30:34
From: Keeley
Subject: Triangle construction in Euclidean Geometry

Given two angles, alpha and beta (angles A and B), and the
semiperimeter, construct the triangle.
```

```
Date: 03/15/2002 at 05:15:32
From: Doctor Floor
Subject: Re: Triangle construction in Euclidean Geometry

Hi, Keeley,

If you can construct the semiperimeter, you can construct the
perimeter as well - you can duplicate by taking a diameter of the
circle with the semiperimeter as radius.

So let's suppose you have a segment IJ with the perimeter of ABC as
length. Take a segment DE, a lot smaller, that is parallel to IJ, and
complete it to triangle DEF using that angle D = alpha and angle E =
beta; both are given. Produce DE to a line, and take the outward
intersections of this line with the circles with centers D and E both
through F. This gives points G and H.

The length of GH is the perimeter of DEF, while the length of IJ is
the perimeter of ABC. Also, IJ and GH are parallel. If we now
intersect lines IG and JH to find point K, then IJK and GHK are
similar triangles, and K is the center of similitude. So we can use K
to find the correct length AB: Intersect KD with IJ and KE with IJ to
find A and B respectively.

Now you can finish!

If you have more questions, just write 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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