Question is the title. I know that this is the case, but I don't know why. You can take two points that are at the same distance from an axis but whose line doesn't form a 90° angle to the axis and they wouldn't be symmetrical, so why is this?
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Take points A(-2,4) and B(1,5) and point C(0,4), the straight line distance value from both points to C is 2, yet A and B are not symmetrical. – Jose Feb 07 '18 at 20:32
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I don't agree because it's the same line i.e. the same axis, regardless though, my point remains because central symmetry (point symmetry) also forms a perpendicular line to the reflection line. – Jose Feb 07 '18 at 20:45
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First of all, $AC\ne BC$ in your example. You don’t have central symmetry, nor can symmetry w/r to a line be judged without having specified the line. More importantly, though, what’s your definition of reflection in a line in the first place? Without that, we’re left with trying to read your mind as to where your misunderstanding or doubts might lie. – amd Feb 08 '18 at 07:27
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Let $A,B$ be symmetric with respect to line $\ell$. Then each point on $\ell$ is at equal distance to $A$ and $B$, so $\ell$ is bisector of segment $AB$ and this is well known perpendicular to $AB$.
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