Difference between revisions of "Inradius"
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== Problems == | == Problems == | ||
− | *Verify the inequality <math>2r | + | *Verify the inequality <math>R \geq 2r</math>. |
*Verify the identity <math>\cos{A}+\cos{B}+\cos{C}=\frac{r+R}{R}</math> (see [[Carnot's Theorem]]). | *Verify the identity <math>\cos{A}+\cos{B}+\cos{C}=\frac{r+R}{R}</math> (see [[Carnot's Theorem]]). | ||
− | * | + | *[[2007 AIME II Problems/Problem 15]] |
{{stub}} | {{stub}} | ||
[[Category:Geometry]] | [[Category:Geometry]] |
Revision as of 15:55, 22 November 2016
The inradius of a polygon is the radius of its incircle (assuming an incircle exists). It is commonly denoted .
Properties
- If has inradius and semi-perimeter , then the area of is . This formula holds true for other polygons if the incircle exists.
- The in radius 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).
- 2007 AIME II Problems/Problem 15
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