Inradius
The inradius of a polygon is the radius of its incircle (assuming an incircle exists). It is commonly denoted .
![[asy] pathpen = linewidth(0.7); pair A=(0,0),B=(4,0),C=(1.5,2),I=incenter(A,B,C),F=foot(I,A,B); D(MP("A",A)--MP("B",B)--MP("C",C,N)--cycle); D(CR(D(MP("I",I,SW)),inradius(A,B,C))); D(F--I--foot(I,B,C)--I--foot(I,C,A)); D(rightanglemark(I,F,B,5)); MP("r",(F+I)/2,E); [/asy]](http://latex.artofproblemsolving.com/3/a/d/3ad1b30f1350fc22f90ce7a79f2567025d4fe4c0.png)
Properties
- If
has inradius
and semi-perimeter
, then the area of
is
. This formula holds true for other polygons if the incircle exists.
- The inradius satisfies the inequality
, where
is the circumradius (see below).
- If
has inradius
and circumradius
, then
.
Problems
- Verify the inequality
.
- Verify the identity
(see Carnot's Theorem).
- Special:WhatLinksHere/Inradius: 2007 AIME II Problems/Problem 15
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