Three Points Theorem

(Three points theorem) For a curve γ, the set { {q₁, q₂} | q₁, q₂∈γ, q₁+q₂+q₃= 0 } for a given q₃∈γ is equal to the set { {q, q*} |q∈γ∩γ'} where γ' = {-q-q₃ | q∈γ} is an inversion and parallel translation of the curve γ, and q* =−q−q₃.

The theorem states that for a curve γ and for a given q₃ ∈γ, if there is a pair q₁, q₂ ∈γ that satisfy q₁+q₂+q₃= 0 then the points q₁ and q₂ should be the cross points of γ and γ'. In the following animations, black curve represents γ and red γ'.

(All animations above made of 720 frames per period and each frame spends 40msec.)