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Radical Circle of Three Circles
The radical circle of three circles has center at the radical center
and is orthogonal to each of the three circles.
Let the centers of the circles form the triangle
and let its side lengths be
,
,
.
Using Conway triangle notation, define
,
,
.
Half the triangle's area
is then given by
.
In those terms, suppose the Cartesian coordinates of three centers are at
,
,
and the corresponding radii are
,
,
.
Then the radical circle of the three circles
,
,
is the circle
, where
,
.
Contributed by:
Minh Trinh Xuan
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Exercise: verify that the side lengths are indeed
,
,
from the coordinates of the vertices.
RELATED LINKS
Mapping a 3D Point to a 2D Triangle
(
Wolfram Demonstrations Project
)
Seven Points with Integral Distances
(
Wolfram Demonstrations Project
)
The Sum of the Trilinear Coordinates of a Point
(
Wolfram Demonstrations Project
)
Relating Trilinear and Tripolar Coordinates for a Triangle
(
Wolfram Demonstrations Project
)
Excircles Radical Circle
(
Wolfram
MathWorld
)
Radical Circle
(
Wolfram
MathWorld
)
PERMANENT CITATION
Minh Trinh Xuan
"
Radical Circle of Three Circles
"
http://demonstrations.wolfram.com/RadicalCircleOfThreeCircles/
Wolfram Demonstrations Project
Published: May 10, 2022
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