Difference between revisions of "Circumradius"
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==Euler's Theorem for a Triangle== | ==Euler's Theorem for a Triangle== | ||
− | Let <math>\triangle ABC</math> have circumcenter <math>O</math> and incenter <math>I</math>.Then <cmath>OI=R(R-2r) \implies R \geq 2r</cmath> | + | Let <math>\triangle ABC</math> have circumcenter <math>O</math> and incenter <math>I</math>.Then <cmath>OI^2=R(R-2r) \implies R \geq 2r</cmath> |
==Proof== | ==Proof== |
Revision as of 16:48, 8 June 2017
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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.
Contents
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 circumradius of the triangle is simply
. Also,
Proof
We let ,
,
,
, and
. We know that
is a right angle because
is the diameter. Also,
because they both subtend arc
. Therefore,
by AA similarity, so we have
or
However, remember that area
, so
. Substituting this in gives us
and then bash through algebra to get
and we are done.
--Nosaj 19:39, 7 December 2014 (EST)
Formula for Circumradius
Where
is the Circumradius,
is the inradius, and
,
, and
are the respective sides of the triangle and
is the semiperimeter. Note that this is similar to the previously mentioned formula; the reason being that
.
Euler's Theorem for a Triangle
Let have circumcenter
and incenter
.Then