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A circle is cut into 8 parts, each part has the angle 45 degrees from an arbitrary point. how to prove that the white area is the same as the Gray area? Circle cut into 8 parts

I just want any hint for solving this question. how can I prove this?


marked as duplicate by Martin R, Xander Henderson, Hayk, Blue geometry May 22 at 13:51

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Carter and Wagon's proof without words:

enter image description here

(Image author: Christian Lawson-Perfect, source)

  • $\begingroup$ Thanks for Help. $\endgroup$ – Pankaj Solanki May 22 at 8:53

This theorem is known as the Pizza theorem (see https://en.wikipedia.org/wiki/Pizza_theorem for further reading). There exist algebraic proofs, but the most elegant proof for the 8 segment case is below: enter image description here

In general, alternating areas are equal iff the number of segments is a multiple of 4 at least 8.

  • $\begingroup$ How are segments counted? If this is eight from the concurrence point, then four will not work. $\endgroup$ – Oscar Lanzi May 22 at 13:44
  • $\begingroup$ @OscarLanzi Sorry about that, I've fixed my answer. $\endgroup$ – auscrypt May 22 at 13:46

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