Let’s have some fun with the Pythagorean theorem, we’ll deduce the general equation of circle in **R**^{2} with the Pythagorean theorem.

The Pythagorean theorem states that right triangle’s sum of squares of the other sides equal to the square of the hypotenuse: *a*^{2} + *b*^{2} = *c*^{2}.

The definition of the (analytical) circle: The set of the points that’s distance from a given point *P* is constant *r*.

In the picture below is drawn to plane to arbitrary a position a circle, that’s center is *P* = (*x*_{0}, *y*_{0}). Inside the circle is drawn a right triangle that’s other side’s end point is (*x*, *y*‘) and the other’s endpoint is (*x*‘, *y*).

Now the horizontal side’s length is |*x* – *x*_{0}| and vertical sides’s length is |*y* – *y*_{0}| and hypotenuse again is *r*, when according to Pythagorean theorem

(|*x* – *x*_{0}|)^{2} + (|*y* – *y*_{0}|)^{2} = *r*^{2}

<=> (*x* – *x*_{0})^{2} + (*y* – *y*_{0})^{2} = *r*^{2}

Because square is positive, the absolute values can be removed and what’s remaining is the familiar general equation of circle.

The simplest case is the circle where the center is at origin.

Then the length of sides of the right triangle are

|*x* – 0| = |*x*| and |*y* – 0| = |*y*|

and the hypotenuse is again *r*, when according to Pythagorean theorem

*x*^{2} + *y*^{2} = *r*^{2}

I don’t know how ”official” way to deduce the equation of circle this is, but at least this gives the result… 🙂

There is story behind all this that began in the beginning of the year 1999…

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Posted by Markus in Mathematics Tags: Mathematics, Pythagorean Theorem