Triangle Construction Given Two Angles and SemiperimeterDate: 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/ |
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