Given coordinates of $C$ and $\overline{AC} = \overline{BC}$, find $A$ and $B$.

Question

If $C$ has coordinates $(\sqrt 7, \sqrt3)$ and $\overline{AC} = \overline{BC}$, what are the rational coordinates of $A$ and $B$?

Answer 1

Let the rational coordinates of $a,b$ be $(x_a,y_a),(x_b,y_b)$ respectively

We need $$(x_a-\sqrt7)^2+(y_a-\sqrt3)^2=(x_b-\sqrt7)^2+(y_b-\sqrt3)^2$$

$$\implies x_a^2+y_a^2-2\sqrt7x_a-2\sqrt3y_a=x_b^2+y_b^2-2\sqrt7x_b-2\sqrt3y_b$$

Equating the irrational parts, $$-2(\sqrt7x_a+\sqrt3y_a)=-2(\sqrt7x_b+\sqrt3y_b)$$

$$\implies \sqrt7(x_a-x_b)=\sqrt3(y_b-y_a)$$

As $x_a,y_a,x_b,y_b$ are rational ,the above equality holds iff $x_a-x_b=y_b-y_a=0$

So, $a,b$ can not be distinct if they have rational coordinates