Tangent as a Radical Axis

The applet below illustrates Problem 4 from the 1995 Asia Pacific Mathematical Olympiad:

In the applet, T is an arbitrary point in the plane of the cyclic quadrilateral PQRS. The applet shows circumcircle of triangles PQT and RST. The configuration of interest is where the two circles touch at their common point. On such occasions T is renamed to I.

Solution.

Copyright © 1996-2010 Alexander Bogomolny

Let C be the circumcircle of the quadrilateral PQRS and C' and C'' the circumcircles of triangle PQT and RST, respectively. PQ is the radical axis of C and C'; RS is the radical axis of C and C''. Point J is the radical center of the three circles. It follows that the radical axis of C' and C'' passes through J. On the other hand, the radical axis of two touching circles is their common tangent through the point of tangency. Therefore, the tangent to the two circles C' and C'' at I passes through J.

According to the Intersecting Chords Theorem, IJ² = PJ×QJ (and also, of course, IJ² = RJ×SJ.) For the fixed P, Q (or R, S), the product on the right is fixed, implying that IJ is constant. This means that I is located on a circle centered at J of radius √PJ×QJ.

When I lies on PQ, circle IPQ degenerates into a straight line, a circle of infinite radius. Similarly, when I lies on RS, circle IRS becomes a straight line. When I is inside C, C' and C'' touch externally; otherwise they touch internally.

(Elsewhere Vo Duc Dien offers a more detailed solution.)

Radical Axis and Radical Center

• A Property of the Line IO: A Proof From The Book
• Cherchez le quadrilatère 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

Copyright © 1996-2010 Alexander Bogomolny