Inner polygon (1): Any Polygon contains an inner positive triangle.

The fifth chapter of this book is very good B. It describes a series of Theorem related to the inner graph of polygon and its proof. Interestingly, it is also a research on the interior drawing of polygon. When the research objects are different, the methods of proof are also brilliant, and it is very rare that, these proofs are of great value. After reading these clever proofs, I can't wait to share them with you. Here, let's warm up and take a look at the simplest situation: Can a polygon always have an equilateral triangle in it.

 
 
 

The answer is yes. Any Polygon always has an inner equi-edge triangle. A very intuitive proof is that p is a point on the boundary of a polygon and Q is a dynamic point on the polygon. Use PQ as an equi triangle and record the third vertex of this triangle as R. When q is sufficiently close to P, R is obviously inside the polygon; when Q moves to Q, the distance from any point in the polygon to P is smaller than that of PQ, so the R point is only outside the polygon. But the Motion Track of R is obviously continuous, so it must pass through the boundary of the polygon during the motion process. At this point, we find the three points P, Q, and R on the boundary of the polygon, which form an equi-edge triangle.

 
 
 

Another beautiful proof is shown in. So that p is a point on the side of a polygon, And the whole polygon is rotated 60 degrees clockwise around the P point. Obviously, after the line segment where P points are located is rotated, some will fall into the original polygon, and some will fall outside the original polygon. Therefore, the rotated polygon must have other intersections with the boundary of the original polygon. Otherwise, its side cannot form a closed loop. Set another intersection to Q, and record the point obtained after the Q point rotates around P 60 degrees counterclockwise as the R point. Obviously, the R point is on the original polygon, and △pqr is an equi-edge triangle.

 
Note: many of these series of proofs are incomplete, and many seemingly obvious details need further proof. However, this still does not affect our appreciation of these elegant proofs (especially their ideas ).