### Tangent as a Radical Axis

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

Let PQRS be a cyclic quadrilateral such that the segments PQ and RS are not parallel. Consider the set of circles through P and Q, and the set of circles through R and S. Determine the set I of points of tangency of circles of these two sets. |

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.

What if applet does not run? |

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### Tangent as a Radical Axis

What if applet does not run? |

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,

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

- 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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