Why Does Time Remain Constant on Any Chord in Galileo's Circle?

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Why is it that if you have a circle. If you put any rod from the top center of the circle to any other point on the circle (this would be defined as a chord), then why, if the bead starts at the top, is the time required to slide down any chord independent of the particular chord chosen?

I have no idea how to even start this... any tips would be appreciated.
 
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You know how to find the time taken to slide down any given ramp, given length and height of the ramp, right? So write that down. Now put in the relation between length and height if the ramp is a chord on the circle.
 
What a nice problem -- I'd never heard of this one before! I'll just chime into say that you may find the trig identity for sin(theta/2) useful near the end...
 

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