MATHEMATICS AND ORIGAMI

Mathematics and Origami

8.2.8.4 INCENTER
Is the intersection of the three bisectors of a triangle.

1- Fold the three bisectors to get I.

2- Equal distances DE and DF from a point D of a bisector BX, to corresponding sides.

(∆DEB and ∆DBF are equal: right angled with equal angles in B and common side BD).

3- Hence, equal distances from I to the three sides. One of the most important geometric prop-

erties of origami is that radii IA 1 IB 1 IC 1 I can be folded from the incenter to the respective

sides.

4- FUSHIMI ́S THEOREM OF INCENTER: Any triangle folded through its incenter up to

their vertices and along a radius such as IB 1 of fig. 3, becomes a flattened figure.

5- There it is the resultant figure.

6- Is an enlargement of 5 to justify, together with 7, the flattening process. Angles 1, 2, 3, 4

are taken in the order they have been produced, so their sum will add up to zero, for the end

falls over the beginning:

Angs. 1 –2 + 3 – 4 = 0

Angs. 1 + 3 = 2 + 4

7- Unfolding fig. 6, it is:

Angs. 1 + 2 + 3 + 4 = 360º

Angs. (1 + 3) + (2 + 4) = 360º

Angs. 1 + 3 = 180º ; Angs. 2 + 4 = 180º

8.2.8.5 RUMPLED AND FLATTENED ORIGAMI
The first thing to ask is if this matter has anything at all to do with triangle singular points. The
answer is yes: it is based upon Fushimi ́s theorem.

FLATTENIG CONDITION OF A FOLDED FIGURE: To fold flat a figure around a node it is

a necessary condition (but not sufficient) that the angles having their vertices on the node and

their sides being the corresponding fold lines, are supplementary taken in alternate order. The

other condition is that, when rumpling or flattening is produced, the paper will not interfere

within itself.

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