Radical Center: What Is It About?
A Mathematical Droodle

Explanation

Copyright © 1996-2008 Alexander Bogomolny

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.

1. 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.

2. Two circles with centers O₁ and O₂ are tangent at point T. Through a point P on their common tangent draw two lines PS₁ and PS₁. Assume this creates the intersection points A₁, B₁, and A₂, B₂, respectively. Then the four points A₁, B₁, B₂, A₁ are cyclic.

3. The applet may also be seen as presenting a solution to a construction problem: Find a circle through two given points A₁, B₁ tangent to a given circle (that with center O₂.)

   One solution is this. Draw any circle through the given points A₁, B₁ and intersecting the given circle. Let this be circle O. The radical axes of the circles O, O₂, and the unknown circle O₁ meet at some point P. The tangent from P to the circle O₂ serves as the radical axis of O₂ and the unknown circle O₁. Therefore, the point of tangency T belongs to the two circles, O₁ in particular. Now the three points A₁, B₁, and T define O₁ uniquely.

References

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

• 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

Copyright © 1996-2008 Alexander Bogomolny