Difference between revisions of "Circumradius"
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==Formula for a Triangle== | ==Formula for a Triangle== | ||
Let <math>a, b</math> and <math>c</math> denote the triangle's three sides, and let <math>A</math> denote the area of the triangle. Then, the measure of the of the circumradius of the triangle is simply <math>\frac{abc}{4A}</math> | Let <math>a, b</math> and <math>c</math> denote the triangle's three sides, and let <math>A</math> denote the area of the triangle. Then, the measure of the of the circumradius of the triangle is simply <math>\frac{abc}{4A}</math> | ||
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| + | ==Euler's Theorem for a Triangle== | ||
| + | Let <math>\triangle ABC</math> have circumradius <math>R</math> and inradius <math>r</math>. Let <math>d</math> be the distance between the circumcenter and the incenter. Then we have <cmath>d^2=R(R-2r)</cmath> | ||
==See also== | ==See also== | ||
Revision as of 20:45, 6 November 2007
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The circumradius of a cyclic polygon is the radius of the cirumscribed circle of that polygon. For a triangle, it is the measure of the radius of the circle that circumscribes the triangle. Since every triangle is cyclic, every triangle has a circumscribed circle, or a circumcircle.
Formula for a Triangle
Let 
and 
denote the triangle's three sides, and let 
denote the area of the triangle. Then, the measure of the of the circumradius of the triangle is simply 
Euler's Theorem for a Triangle
Let 
have circumradius 
and inradius 
. Let 
be the distance between the circumcenter and the incenter. Then we have ![\[d^2=R(R-2r)\]](http://latex.artofproblemsolving.com/7/e/2/7e2fcd01749b32bd1072381de2ac6589dd730703.png)
