MATHEMATICS AND ORIGAMI

Mathematics and Origami

13.3.4 ELLIPSE: PONCELET THEOREM.

This theorem is applicable to all of three conics, but here we shall prove it by means of

paper folding, for the ellipse only, and just for one of the focuses. It can be done alike for the

other one. Its enunciation is as follows (Fig. 1):

Let PT and PT ́ the pair of tangents to ellipse e from the exterior point P:

1- PF ́ is the bisector of Ang. T ́F ́T (a angles are congruent; likewise a ́ are also congru-

ent).

2- Angles T ́PF ́ = b ́ and TPF = b are congruent.

Fig. 2 shows ellipse e and its director circle cd (Point 13.3.2) within a paper rectangle p.

We can also see three fold lines through P: PX along tangent PT ́ (valley fold); PF ́, valley fold

as well, and PC (mountain fold). These lines extend up to the rectangle ́s boundaries.

Fig. 3 is the result of folding flat Fig.2. When unfolding Fig. 3 we get Fig. 4, in which we can

see valley fold PZ that implies these consequences:

• Ang. CPG must measure 180º - Ang. XPF ́ (see Point 8.2.8.5).

1

P

T ́

T

A

F O F ́ A ́ F F ́

P

T ́

C

X

A O

A ́

T

2

p

e cd e

A F O F ́

X

P

T

C

T ́

4

A ́

G

Z

3

p

cd p

eecd

P

X

Z