Pythagorean theorem vs an equation of a circle? Today I told my teacher that the equation of a circle looks like to the Pythagorean theorem to me, but he said that I'm wrong and to re think it.
Why $(x-h)^2 + (y-k)^2 = r^2$ is not a PT, it looks just like PT -- we square two numbers, add them and we get another number squared.
Where $(h,k)$ is the center and $r$ is the radius of a circle.
Can someone explain in more details?
 A: Suppose the center of the circle is (0,0) and $(x,y)$ is a point of the circle of radius $r$, $x^2+y^2=r^2$
The points $A=(0,0), B=(y,0), C=(x,y)$ define a rectangle triangle. 
A: I can NOT believe you teacher said that!  Take a circle with radius r centered at (h,k).  Pick an arbitrary (x,y) on the circle.  Drop a vertical line down from (x,y) to (x, k).  Slide a horizontal line from (h,k) to (x,k).  These three points (x,y) to (x,k) to (h,k) back to (x,y) form a triangle.  Not just a triangle a right triangle.  
So the triangle has base side A = (h,k) to (x,k).  It has length (x - h).
The triangle has height side B = (x,k) to (x, y).  It has length (y - k).
The triangle has hypothenuse C = (h,k) to (x,y). It has lenght r.
The pythagorean theorem says $A^2 + B^2 = C^2$ or $(x-h)^2 + (y - k)^2 = r^2$.  This gives us a formula for any point of the circle that relates the x value to its y value.
It is base entirely upon the Pythagorean Theorem.
A: When you write the equation of a circle, you start from a definition as:

The locus of points $P$ that have the same distance $r$ from a fixed
  point $C$ (the center).

Then you write the equation as $\overline PO=r$ using the standard definition of (Pythagorean) distance for points represented in an orthogonal reference system:
$$
\overline PO=\sqrt{(x_P-x_O)^2+(y_P-y_O)^2}=r
$$
and squaring you find the equation
$$
(x_P-x_O)^2+(y_P-y_O)^2=r^2
$$
Clearly, this seems the classical pythagorean formula, and a figure confirm that there is a rectangular triangle with $r$ as hypotenuse.
But note that this is a consequence of many other assumptions: the definition of the circle, the use of a particular distance, an the use of an orthogonal reference system.  And, finally, your equation refers to an infinite set of points $P$ not to a single triangle.
So, in my opinion, your teacher is right in inviting you to better think to the question and to the difference between a theorem and a consequence of this theorem.
In a different coordinate system, the equation of a circle can be very different, but the PT is always the same if the metric (the distance) is Pythagorean.
