I thought I had an analytic solution but realized it was wrong after I posted the problem. My apologies!
Yes I was just thinking about how the answer was similar! I’m going to go back and look into that a little. I’ll post a note and update the post if I find anything.
I thought I had an analytic solution but realized it was wrong after I posted the problem. My apologies!
Yes I was just thinking about how the answer was similar! I’m going to go back and look into that a little. I’ll post a note and update the post if I find anything.
Ah... so I wasn't the only one that couldn't get the integral to work :)
Given how close it was to the chord paradox results I figured I had missed a trick or something
Please see the proofs at the following URL:
https://drive.google.com/file/d/1Rn-hWJ0NmgsBVxH5pvcWQrQ2POcE_Xod/view?usp=sharing
This resource contains proofs of the following results:
1. The x-intercept of the line passing through (cos a₁, sin a₁) and (cos a₂, –sin a₂) is given by
x = cos((a₁ + a₂) / 2) / cos((a₁ – a₂) / 2).
2. The average value of (1 – |x|) / (1 + |x|) is 1/3.
3. The double integral
2 ∫₀^π ∫₀^π [1 – (sin((a₂ – a₁)/2) / sin((a₁ + a₂)/2))] / [1 + (sin((a₂ – a₁)/2) / sin((a₁ + a₂)/2))] da₁ da₂
equals π²/3.
4. The double integral
8 ∫₀^(π/2) ∫₀ᵘ [1 – (sin u / sin v)] / [1 + (sin u / sin v)] dv du
equals –π²/3.