Isosceles on the Sides of a Triangle: What is this about?
A Mathematical Droodle

Explanation

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The proof is straightforward. Consider three circles cD, cE, cF centered at D, E, F with radii DB = DC, EC = EA, and FA = FB, respectively. cD and cE meet in C and some other point, say, c so that Cc is their common chord and serves therefore their radical axis. The chord Cc is orthogonal to the line of centers, DE. That is, Cc coincides with the perpendicular from C to DE. Similarly, the other two perpendiculars, serve as the radical axes of the other two pairs of the circles. But the radical axes of three circles concur in a point, the radical center of the circles. So do the perpendiculars at hand.

References

1. R. Honsberger, Mathematical Delights, MAA, 2004, pp. 19-20

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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