### Two Circles on a Side of a Triangle: What is this about?

A Mathematical Droodle

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Copyright © 1996-2018 Alexander BogomolnyThe applet may suggest the following statement:

Let point P lie on a side AB of ΔABC. Circle C(A) passes through A and P and is tangent to AC, circle C(B) passes through B and P and is tangent to BC. Q is the second point of intersection of C(A) and C(B). Show that, regardless of the position of P on AB, line PQ passes through a fixed point X. |

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### Proof

First note that in C(A)

∠AQP = ∠CAP. |

Similarly, in C(B),

∠BQP = ∠CBP. |

If so, then

∠AQB + ∠ACB = 180°, |

which says that quadrilateral ACBQ is cyclic so that Q lies on the circumcircle of ΔABC.

Let X be the point of intersection of PQ with the circumcircle. Inscribed angles AQX and BAC are equal. Therefore, the arcs AX and BC they subtend are also equal. It follows that CX is parallel to AB. Hence, X is independent of P.

### 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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