Crux’s Crux’s Crux

Summary

A result of Archimedes states that for perpendicular chords passing through a point P in the interior of the unit circle, the sum of the squares of the lengths of the chord segments from P to the circle is equal to 4. A generalization of this result to $n\ge 2$ chords is given. This is done in the backdrop of revisiting Problem 1325 from Crux Mathematicorum, for which a new solution is presented.

MSC:

• Primary 51M04
• Secondary 00A08

Acknowledgment

The author is grateful to the reviewer and the Editor for useful suggestions on improving the exposition.

Notes

1 First ‘Crux’ in the title.

2 Second ‘crux’ in the title.

3 Third ‘crux’ in the title.

Additional information

Notes on contributors

Amol Sasane

AMOL SASANE earned a Bachelors Degree in Electrical Engineering from the Indian Institute of Technology, Mumbai, a PhD in Mathematics from the University of Groningen, the Netherlands, and a Masters in Theoretical Physics from Lund University, Sweden. He is now a professor of mathematics at the London School of Economics. His research interests lie in the field of applicable analysis.