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three conics theorem
The three radical axes meet in a single point, the radical center, for the following reason. The radical axis of a pair of circles is defined as the set of points that have equal power with respect to both circles. For example, for every point on the radical axis of circles 1 and 2, the powers to each circle are equal: . Similarly, for every point on the radical axis of circles 2 and 3, the powers must be equal, . Therefore, at the intersection point of these two lines, all three powers must be equal, . Since this implies that , this point must also lie on the radical axis of circles 1 and 3. Hence, all three radical axes pass through the same point, the radical center. The radical center has several applications in geometry. It has an important role in a solution to Apollonius' problem published by Joseph Diaz Gergonne in 1814. In the
Radical Center
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Radical Axis and Center
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In geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, the power center of three circle
A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
s, also called the radical center, is the intersection point of the three radical axes of the pairs of circles. If the radical center lies outside of all three circles, then it is the center of the unique circle (the radical circle) that intersects the three given circles orthogonally; the construction of this orthogonal circle corresponds to Monge's problem. This is a special case of ththree conics theorem
The three radical axes meet in a single point, the radical center, for the following reason. The radical axis of a pair of circles is defined as the set of points that have equal power with respect to both circles. For example, for every point on the radical axis of circles 1 and 2, the powers to each circle are equal: . Similarly, for every point on the radical axis of circles 2 and 3, the powers must be equal, . Therefore, at the intersection point of these two lines, all three powers must be equal, . Since this implies that , this point must also lie on the radical axis of circles 1 and 3. Hence, all three radical axes pass through the same point, the radical center. The radical center has several applications in geometry. It has an important role in a solution to Apollonius' problem published by Joseph Diaz Gergonne in 1814. In the
power diagram
In computational geometry, a power diagram, also called a Laguerre–Voronoi diagram, Dirichlet cell complex, radical Voronoi tesselation or a sectional Dirichlet tesselation, is a partition of the Euclidean plane into polygonal cells defined from ...
of a system of circles, all of the vertices of the diagram are located at radical centers of triples of circles. The Spieker center of a triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
is the radical center of its excircles. Several types of radical circles have been defined as well, such as the radical circle of the Lucas circles.
Notes
Further reading
* * * * * *External links
* * *{{mathworld, urlname=MongesProblem, title=Monge's problemRadical Center
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Cut-the-Knot
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet Union, Soviet-born Israeli Americans, Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow ...
Radical Axis and Center
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Cut-the-Knot
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet Union, Soviet-born Israeli Americans, Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow ...
Elementary geometry
Geometric centers