How can i Prove that the gray area is the same as white area?

Question

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?

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

The area of a triangle with respect to given two sides $a,b$ and angle between the sides is $0.5 a b \sin \theta$ s.t $\theta$ is the angle between the two sides can you proceed ? — IrbidMath, May 22 '19 at 07:56
This is the Pizza Theorem. See here: https://en.wikipedia.org/wiki/Pizza_theorem — Christian Blatter, May 22 '19 at 08:26
See also: https://math.stackexchange.com/q/581895/42969, https://math.stackexchange.com/q/865818/42969 — Martin R, May 22 '19 at 11:13

score 6 · Accepted Answer · answered May 22 '19 at 08:47

6

Carter and Wagon's proof without words:

(Image author: Christian Lawson-Perfect, source)

answered May 22 '19 at 08:47

Saša

15,906

Thanks for Help. – Pankaj Solanki May 22 '19 at 08:53

auscrypt · Answer 2 · 2019-05-22T13:46:06.727

5

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:

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

edited May 22 '19 at 13:46

answered May 22 '19 at 08:50

auscrypt

8,186

How are segments counted? If this is eight from the concurrence point, then four will not work. – Oscar Lanzi May 22 '19 at 13:44
@OscarLanzi Sorry about that, I've fixed my answer. – auscrypt May 22 '19 at 13:46

How can i Prove that the gray area is the same as white area?

2 Answers2