# Diametrically opposite points go to diametrically opposite points under stereographic projection

I asked this question before here but I didn’t get a proper answer. So here I am stating it more clearly :

Suppose $P_1$ and $P_2$ are two diametrically opposite points of a circle $C$ in the complex plane and suppose $\sigma (z)$ denotes the image of $z\in\mathbb{C}$ on Riemann sphere due to the inverse of the stereographic projection. Then I need to prove that $\sigma (P_1)$ and $\sigma (P_2)$ are also diametrically opposite points of the circle $\sigma (C)$.

Intuitively this is obvious but I need to prove this fact algebraically (i.e. using coordinate geometry ).

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You should include a link to the earlier question. –  Gerry Myerson Aug 20 '12 at 13:37
@GerryMyerson, link included. –  pritam Aug 21 '12 at 6:48
Cross-posted to MO where it has been observed that the statement, though possibly "intuitively obvious", is false. –  user16299 Aug 23 '12 at 7:28