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I have a question regarding how you go about constructing a proof of an idea. I'm not familiar with very many examples - Pythagoras's proof, Euclid's proof that $\sqrt{2}$ is irrational, and a handful of others I know.

I'm trying to 'prove' something that seems intuitively true, but I know that's not enough in the world of maths.

So, take $2$ coordinates on an $x, y$ graph and join them with a line. Take the midpoint of that line and construct a perpendicular line. Taking the centre of a circle to be anywhere along this line is there an infinite number of circles that pass through our original coordinates.

$(x−h)^2+(y−k)^2=r^2$ describes a generic circle and if $(h, k)$ is the center of the circle.

So, as I've said this seems intuitively true and I'm able to show that it is true for certain circles, but how do I extend that into being true for an infinite number of circles?

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You can note that the location of the two points is irrelevant, as is the actual distance between them, So if you feel like it you can choose $x$-axis, $y$-axis, unit of distance so that our two points are $(-1,0)$ and $(1,0)$. – André Nicolas Jun 5 '13 at 14:58
Call the first two points $A$, $B$. Let $P$ be any point on your line (there are infinitely many such points). Use Pythagoras to show that the distances $|PA|$ and $|PB|$ are equal. By definition of a circle, $A$ and $B$ lie on the circle with center at $P$ and radius $|PA|$. – Erick Wong Jun 5 '13 at 15:01
You have congruent right triangles with common base and equal sides, so the hypotenuse is the circle's radius. Circle's center is the point where base meets hypotenuse. As base heads toward infinity, hypotenuse and radius head toward infinity. – Fred Kline Jun 5 '13 at 15:06

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