Pinning Butterfly on Radical Axes

The applet below illustrates a solution to the traditional Butterfly Problem was communicated to me by Giles Gardam, a member of Australia's 2007 and 2008 IMO teams, and is due to his mentor, Ivan Guo, who won a gold medal at IMO 2004.

In a circle T, two chords AB and CD intersect at the midpoint of a third chord PQ. AD meets PQ in X, BC meets PQ in Y. The task is to show that M is also the midpoint of XY.

Reflect B and C and the circle T in M (by reflect in a point I mean a dilation by factor -1 about that point). B', C', and T' be the reflections of B, C and circle T, respectively. Note that as M is the midpoint of PQ, T' passes through P and Q.

By power of point M with respect to T, AM×BM=CM×DM.

Thus AM×B'M=C'M×DM, so by converse of power of a point, AB'C'D is a cyclic quadrilateral. Let its circumcircle be T''.

We now consider the three radical axes of the three circles, which are PQ, AD and B'C', and concur by the radical axes theorem. Thus B'C' intersects PQ at X. The reflections of BC and PQ are B'C' and PQ, so considering their intersections, we have that X is the reflection of Y, thus M is the midpoint of XY.

Butterfly Theorem and Variants

1. Butterfly theorem
2. 2N-Wing Butterfly Theorem
3. Better Butterfly Theorem
4. The Lepidoptera of the Circles
5. The Lepidoptera of the Quadrilateral
6. The Lepidoptera of the Quadrilateral II
7. Butterflies in Ellipse
8. Butterflies in Hyperbola
9. Butterflies in Quadrilaterals and Elsewhere
10. Pinning Butterfly on Radical Axes
11. Shearing Butterflies in Quadrilaterals
12. The Plain Butterfly Theorem
13. Two Butterflies Theorem
14. Two Butterflies Theorem II
15. Two Butterflies Theorem III
16. Algebraic proof of the theorem of butterflies in quadrilaterals
17. William Wallace's Proof of the Butterfly Theorem
18. Butterfly theorem, a Projective Proof

Radical Axis and Radical Center

• 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

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Copyright © 1996-2012 Alexander Bogomolny