Reflection: Line about a Circle

Author:
Ryan Hirst
Theorem without proof: The image of a line reflected across the arc of a circle, is a circle. The image of a ray or segment is a circular arc.
GeoGebra Applet
I used algebra for the proof. Having guessed (observation) that the reflection was a circle, I set up the vector problem: There is a single point M and a corresponding constant c which satisfy, for any point X on the line, X' = M + c (cos θ, sin θ) The crux is this. Drop an altitude from A to the line, and draw the intersection P. Reflect P about α: On the reflection, P' is the closest point to the given line. Now consider a second point Z. very far from A. The further Z gets, the closer Z' is to A: has constant length, and r is a constant. The denominator increases without bound. In the limit as Z gets infinitely far away, Z' = A. So both ends of the line project to point A. The figure is closed. A line doesn't make any weird deviations, it just keeps on keepin' on. The reflection should have constant curvature. A closed figure with constant curvature is a circle, and if this is all true, AP' should be a diameter. Test the midpoint M of AP' directly. Success. _________________ The Tangent Circle Problem: Iteration:

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