请输入您要查询的字词:

 

单词 IncircleRadiusDeterminedByPythagoreanTriple
释义

incircle radius determined by Pythagorean triple


If the sides of a right triangleMathworldPlanetmath are integers, then so is the radius of the incircleMathworldPlanetmath of this triangle.
For example, the incircle radius of the Egyptian triangle is 1.

Proof.  The sides of such a right triangle may be expressed by the integer parametres m,n with m>n>0  as

a= 2mn,b=m2-n2,c=m2+n2;(1)

the radius of the incircle (http://planetmath.org/Incircle) is

r=2Aa+b+c,(2)

where A is the area of the triangle.  Using (1) and (2) we obtain

r=22mn(m2-n2)/22mn+(m2-n2)+(m2+n2)=2mn(m+n)(m-n)2m(m+n)=(m-n)n,

which is a positive integer.

Remark.  The corresponding radius of the circumcircleMathworldPlanetmath need not to be integer, since by Thales’ theorem, the radius is always half of the hypotenuseMathworldPlanetmath which may be odd (e.g. 5).

随便看

 

数学辞典收录了18232条数学词条,基本涵盖了常用数学知识及数学英语单词词组的翻译及用法,是数学学习的有利工具。

 

Copyright © 2000-2023 Newdu.com.com All Rights Reserved
更新时间:2025/5/4 3:35:45