(Three points theorem) For a curve γ, the set { {q_{1}, q_{2}} | q_{1}, q_{2}∈γ, q_{1}+q_{2}+q_{3}= 0 } for a given q_{3}∈γ is equal to the set { {q, q*} |q∈γ∩γ'} where γ' = {-q-q_{3} | q∈γ} is an inversion and parallel translation of the curve γ, and q* =−q−q_{3}.

The theorem states that for a curve γ and for a given q_{3} ∈γ, if there is a pair q_{1}, q_{2} ∈γ that satisfy q_{1}+q_{2}+q_{3}= 0 then the points q_{1} and q_{2} should be the cross points of γ and γ'. In the following animations, black
curve represents γ and red γ'.