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In geometry, the power center of three circles, 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 th
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 ''h'' with respect to both circles. For example, for every point P on the radical axis of circles 1 and 2, the powers to each circle are equal, ''h''1 = ''h''2. Similarly, for every point on the radical axis of circles 2 and 3, the powers must be equal, ''h''2 = ''h''3. Therefore, at the intersection point of these two lines, all three powers must be equal, ''h''1 = ''h''2 = ''h''3. Since this implies that ''h''1 = ''h''3, 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
Radical Center
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Radical Axis and Center
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In geometry, the power center of three circles, 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 ''h'' with respect to both circles. For example, for every point P on the radical axis of circles 1 and 2, the powers to each circle are equal, ''h''1 = ''h''2. Similarly, for every point on the radical axis of circles 2 and 3, the powers must be equal, ''h''2 = ''h''3. Therefore, at the intersection point of these two lines, all three powers must be equal, ''h''1 = ''h''2 = ''h''3. Since this implies that ''h''1 = ''h''3, 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
In Euclidean plane geometry, Apollonius's problem is to construct circles that are tangent to three given circles in a plane (Figure 1). Apollonius of Perga (c. 262 190 BC) posed and solved this famous problem in his work (', "Tangenci ...
published by Joseph Diaz Gergonne in 1814. In the power diagram 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 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-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...
Radical Axis and Center
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Cut-the-Knot
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...
Elementary geometry
Geometric centers