Let's also talk about the pizza question: I will share some beautiful proofs.

Apparently, the circle center of pizza is divided into four straight lines and a circumference is divided into eight equal portions, then the entire pizza is also divided into eight equal portions. We can also easily think that if four straight lines are made from a point outside the center of the circle, each of the two adjacent lines is clipped with a 45 degree angle, then these eight pizza cakes are obviously different. The time has come to test your intuition: Which one do you think is bigger than the sum of the blue area and the red area?

 
 
 
 
 
 
 
 
 
 
 
 
 

In fact, the blue area is as large as the red area. But please note that this is definitely not clear. The following fact may surprise many people:TwoVertical lines, then the area of red (the color of the circle center) is greater than the blue area. To prove this, we only need to show that the sum of the Red Area and the sum of the blue area are greater than 0. It is easy to see that G-A = D, H-B = E, I-c = F, and D-F = E, so

(D + G + E + H + C)-(A + B + F + I)
= [(G-A) + d] + [(h-B) + E]-[(I-c) + F]
= 2 * d + 2 * E-2 * F
= 2 * (D + E-f)
= 4 * E> 0

 
Today, I saw the pizza theorem at the scientific squirrel meeting. Coincidentally, I recently read many things related to the pizza theorem. I would like to share with you here. As Xiaoyi said, not long ago, Rick Mabry and Paul deiermann proved the following conjecture: to use N equi-Zhou straight lines to cut a circle. When n is an even number greater than or equal to 4, the two colors have the same area. When n = 1, n = 2, and N is divided by more than 4, the color of the center has more area; in other cases (N is greater than 4 and 4 is divided by more than 1), the color of the center has less area. This proof is long and I have never read it myself. However, it is worth mentioning that, in the case of N = 4, people have found a lot of worships.
In 1994, L. Carter and S. Wagon discovered a cut Complement Method Using the geometric software CABRI to prove that the area of the two colors is indeed the same when n = 4. I made a GIF animation that intuitively showed the entire "proof" process. There are several regions that require strict proof, but it is also very easy. I will not go into detail here.

 
 
Using the integral idea, Jörg härterich provides a more elegant proof. The key to the proof lies in the following theorem: there is any point of P in the circle with a radius of R, and two vertical straight lines are formed over P to generate four intersections with the circle, the distance from P to the four intersections is a, B, c, d, then a ^ 2 + B ^ 2 + C ^ 2 + d ^ 2 = 4 * r ^ 2.

It is easy to prove that, by using the quotitionary theorem, there are obviously (c-a) ^ 2 + (B + d) ^ 2 = (2R) ^ 2, likewise (D-B) ^ 2 + (A + C) ^ 2 = (2R) ^ 2. After the two formulas are expanded, the conclusion is displayed.
Now, point P is used as the origin of Polar Coordinates. If the radius of pizza is 1, R (θ) indicates the distance from P to the circumference of pizza in the θ direction. Consider an area as the point of a small slice area. Because the slice area is equal to the square of the radius multiplied by the half of the radian, the red area is equal

This is exactly half the area of the circle.

Although N = 4 has a lot of beautiful proofs, when n is bigger, the problem will become surprisingly complex. Some more mysterious means should be used for analysis, in the past few decades, nobody has made the crash. Attached is the original paper by Rick Mabry and Paul deiermann. If you are interested, you can study it.