## Four Crossing Circles

The following engaging problem has been posted at the CTK Exchange by Bui Quang Tuan and its solution has been posted by Mariano Perez de la Cruz.

Two circles (O_{1}), (O_{2}) intersect each other at A, B. Two circles (O_{a}), (O_{b}) centered at any point O: (O_{a}) passing A, (O_{b}) passing B.
Other than A: (O_{a}) intersects (O_{1}), (O_{2}) at A_{1}, A_{2} respectively.
Other than B: (O_{b}) intersects (O_{1}), (O_{2}) at B_{1}, B_{2} respectively.
R_{a}, R_{b} are radii of (O_{a}), (O_{b}) respectively. |

Please prove: |

1. (O_{a}) cuts lines A_{1}B_{1} and A_{2}B_{2} at two equal segments. Similarly with (O_{b}). |

2. A_{1}A_{2}/B_{1}B_{2} = R_{a}/R_{b} |

What if applet does not run? |

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Copyright © 1996-2018 Alexander Bogomolny

Two circles (O_{1}), (O_{2}) intersect each other at A, B. Two circles (O_{a}), (O_{b}) centered at any point O: (O_{a}) passing A, (O_{b}) passing B.
Other than A: (O_{a}) intersects (O_{1}), (O_{2}) at A_{1}, A_{2} respectively.
Other than B: (O_{b}) intersects (O_{1}), (O_{2}) at B_{1}, B_{2} respectively.
R_{a}, R_{b} are radii of (O_{a}), (O_{b}) respectively. |

Please prove: |

1. (O_{a}) cuts lines A_{1}B_{1} and A_{2}B_{2} at two equal segments. Similarly with (O_{b}). |

2. A_{1}A_{2}/B_{1}B_{2} = R_{a}/R_{b} |

Observe that segment A_{1}B_{1} is rotated around O, the common center of the two concentric circles (O_{a}), (O_{b}) into AB. AB in turn is rotated around O into A_{2}B_{2}. The product of two rotations is a rotation that maps A_{1}B_{1} onto A_{2}B_{2}.

What if applet does not run? |

The implication is twofold. First of all, the lines of A_{1}B_{1} and A_{2}B_{2} are at the same distance from O, implying that the chords cut off on these lines by the circle (O_{a}) are equal. The same of course holds for the circle (O_{b}). Secondly, the distance between a point and its image under a rotation is proportional to its distance from the center of rotation. Thus _{1}A_{2}/B_{1}B_{2} = R_{a}/R_{b}.

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Copyright © 1996-2018 Alexander Bogomolny

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