MATHEMATICS AND ORIGAMI

Jesús de la Peña Hernández

• New point G lies on T after flattening because:
  
  
  • C → F (symmetry about tangent on T ́; Fig.1 Point 13.3.2)
  
  
  • CP → PF (same symmetry)
  
  
  • F ́G + GC = 2a (radius of cd) = F ́T + TF (T, point on the ellipse).
  
  
  • Not being so, broken line CGF ́ within Fig. 3 would not straighten along CGF ́ when un-
    folding Fig. 4.
  
  
  
  

The result is: ∆CGP = ∆FTP after coincidence of their vertices: C ≡ F; P ≡ P; G ≡ T.

This means that Ang. TPF = Ang. CPG

Hence,

Ang. TPF = Ang F ́PT ́ = Ang. GPC = 180º - Ang. XPF ́ (*)

Ang. CPX = Ang. XPF (symmetry about the tangent in T ́).

Ang. FPF ́= 180º - (Ang. XPF + Ang. T ́PF ́).

Ang. GPT ́= 180º - (Ang. XPC + Ang. CPG)

Ang. F ́PG = Ang. T ́PG + Ang. F ́PT ́

Ang. F ́PT = Ang. F ́PF + Ang. FPT

Therefore Ang. F ́PG = Ang. F ́PT : it demonstrates the 1st part of Poncelet theorem.

(*) ,, ,, 2nd ,, ,,

Interlude