### Radical Center: What Is It About?

A Mathematical Droodle

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Copyright © 1996-2018 Alexander Bogomolny

### Radical Center

The diagram in the applet can be viewed as presenting several problems. Far as *I* can see, all are related to the notion of Radical Center of three circles. Let me know if you see additional interpretations.

What if applet does not run? |

The radical axes of three circles taken by two meet in a point (radical center of the three circles.) The applet presents a particular case where two of the circles are tangent while all three have common points. Indeed, the radical axis of tangent circles is their common tangent. The radical axis of two intersecting circles is the line containing their common chord.

Two circles with centers O

_{1}and O_{2}are tangent at point T. Through a point P on their common tangent draw two lines PS_{1}and PS_{1}. Assume this creates the intersection points A_{1}, B_{1}, and A_{2}, B_{2}, respectively. Then the four points A_{1}, B_{1}, B_{2}, A_{1}are cyclic.The applet may also be seen as presenting a solution to a construction problem: Find a circle through two given points A

_{1}, B_{1}tangent to a given circle (that with center O_{2}.)One solution is this. Draw any circle through the given points A

_{1}, B_{1}and intersecting the given circle. Let this be circle O. The radical axes of the circles O, O_{2}, and the unknown circle O_{1}meet at some point P. The tangent from P to the circle O_{2}serves as the radical axis of O_{2}and the unknown circle O_{1}. Therefore, the point of tangency T belongs to the two circles, O_{1}in particular. Now the three points A_{1}, B_{1}, and T define O_{1}uniquely.

### References

- S. Savchev, T. Andreescu,
*Mathematical Miniatures*, MAA, 2003, pp. 112-113

### Radical Axis and Radical Center

- How to Construct a Radical Axis
- A Property of the Line IO: A Proof From The Book
- Cherchez le quadrilatere cyclique II
- Circles On Cevians
- Circles And Parallels
- Circles through the Orthocenter
- Coaxal Circles Theorem
- Isosceles on the Sides of a Triangle
- Properties of the Circle of Similitude
- Six Concyclic Points
- Radical Axis and Center, an Application
- Radical axis of two circles
- Radical Axis of Circles Inscribed in a Circular Segment
- Radical Center
- Radical center of three circles
- Steiner's porism
- Stereographic Projection and Inversion
- Stereographic Projection and Radical Axes
- Tangent as a Radical Axis
- Two Circles on a Side of a Triangle
- Pinning Butterfly on Radical Axes
- Two Lines - Two Circles
- Two Triples of Concurrent Circles
- Circle Centers on Radical Axes
- Collinearity with the Orthocenter
- Six Circles with Concurrent Pairwise Radical Axes
- Six Concyclic Points on Sides of a Triangle
- Line Through a Center of Similarity

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Copyright © 1996-2018 Alexander Bogomolny

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