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Proof 12: Circle equation

There is a round amphitheatre 1000 metres East and 3500 metres North from your hotel (let’s call those numbers $c_1$ and $c_2$). You go there for a visit.

Wow, it is pretty, when standing in the middle! You wonder what size it has… Your GPS watch can tell this! It can tell you your Eastwards and Northwards coordinates. And Pythagoras’ theorem can then give the radius.

So, you walk due East and then due North until you reach the edge. You read off the coordinates but realise that you forgot to reset the GPS. It was last reset at the hotel, which is now the origin, the reference. So, the GPS tells the total distance East – let’s call it $x$ – and North – let’s call it $y$ from the hotel and not from the theatre centre. The distances $c_1$ and $c_2$ shouldn’t be included. The actual distances are thus $x-c_1$ and $y-c_2$.

Pythagoras’ theorem ($a^2+b^2=c^2$ for a right-angled triangle) now gives us the circle equation:

$$(x-c_1)^2+(y-c_2)^2=r^2$$

Pick any possible $x$ and there is a matching $y$ that makes this equation hold true; meaning, a coordinate set $(x,y)$ on the circle. Run through all possible $x$ values and you get the full circle. We just must only use $x$ values that are inside the circle; any other values are of course impossible. $\quad_\blacksquare$

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