<s>[INST] Rational points on a circle with a center having non rational coordinates Reflection of a rational point with respect to line having its equation with rational coefficient
gives a rational point
What is the number of rational points on a circle with centre $(\sqrt 2, \sqrt 3)$ and with any radius
In this the centre is not rational then also how they decided the answer as 'at most 1'
Original source : https://i.stack.imgur.com/3aMvD.gif
 [/INST] Let $(a,b)$ and $(c,d)$ be distinct rational points on the circle.
$$\begin{array}{rcl}
(a-\sqrt2)^2 + (b-\sqrt3)^2 &=& (c-\sqrt2)^2 + (d-\sqrt3)^2 \\
a^2 + b^2 - 2a\sqrt2 - 2b\sqrt3 &=& c^2 + d^2 - 2c\sqrt2 - 2d\sqrt3\\
a^2 + b^2 - c^2 - d^2 &=& 2(a-c)\sqrt2 + 2(b-d)\sqrt3\\
(a^2 + b^2 - c^2 - d^2)^2 &=& 8(a-c)^2 + 12(b-d)^2 + 8(a-c)(b-d)\sqrt6 \\
\sqrt6 &=& \dfrac{(a^2 + b^2 - c^2 - d^2)^2 - 8(a-c)^2 - 12(b-d)^2}{8(a-c)(b-d)}\\
\end{array}$$
Contradicting the fact that $\sqrt6$ is irrational, unless $a=c$ or $b=d$.

I believe you can deal with the cases:


*

*Case 1: $a=c$ and $b \ne d$

*Case 2: $a \ne c$ and $b=d$


To complete the proof.
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