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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,

Thanks for your question.

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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