Essential theory for origami - Yonekuras theorem

Essential theory for origami - Yonekuras theorem

The Yonekura theorem is not a theorem inspired by rice storage warehouses. It was discovered by Yonekura Ryoko, so it was named Yonekura theorem. After we have seen Hogas first theorem, Hogas second and third theorem, lets learn about another important theorem of paper origami, Yonekuras theorem.

1. Introduction of Yonekang’s theorem:

a. Thinking: Given a right triangle, try to fold a triangle similar to the original triangle.

b. Yonekang theorem:

Introduction: As shown in the figure, in the right triangle ABC, take the midpoint M of BC and make point A coincide with it, so that △MEF is similar to △ABC.

Proof: It is the midpoint on the hypotenuse, MA=MC, it can be deduced that angle 1=angle C, and the four points of AEMF are cocircular, because angle 2=angle 1. Therefore angle 2 = angle C. So △MEF is similar to △ABC.

2. Relevant conclusions drawn from Yonekura’s theorem

a. Tajiri’s theorem:

On the basis of Yonekuras theorem, student Tajiri further discovered that: with M as a fixed point, the angle EMF=90°, and any E and F move freely on AB and AC, the resulting △MEF is similar to △ABC.

b. A more general conclusion (This conclusion was discovered by Mr. Uemura, the teacher of Yonekura and Tajiri)

Thinking: Trying to find in any triangleTo a similar point, the above conclusion is always true.

Steps: Inspired by Ryoko Yonekura’s folding method, Mr. Uemura got:

When M is the circumcenter of △ABC, try to get this conclusion to be true for the circumcenter of any triangle. Prove: △MEF is similar to △ABC

c. Further promotion

Find two creases about the points in the picture above, which leads to the conclusion discovered by student Watanabe:

Conclusion: The relative relationship can be obtained by extending and intersecting the crease lines, and the triangles obtained by folding the vertices are similar to the original triangles.