You start with a perfect square and draw diagonals fron opposite corners. This gives you the center point. And you draw the horizontal and vertical line through that center point, dividing the square into four equal squares, also giving you the first four points on the circle:
And you do that again, dividing the square into 16 equal squares:
Now, here comes the trick that allows you to obtain eight more points on the circle. Draw lines from the corners to the opposite first point of the quarter squares, like image below. The points where these lines cross the first horizontal or vertical are additional points of the circle:
You can do that for all corners:
The beauty of this approach is that you only need a pencil and paper and an ability to draw straight lines, in order to draw ellipses in perspective, because you can divide a square the same way even if the square is projected in perspective, using the vanishing points to draw the horizontals and the verticals:
Now, this is why it works. See image below:
So here's the "difficult math": say this circle has a radius of 1. That means the rectangle is 2 wide in total and divided up into squares with sides (1/2) each. The magenta triangle has a ratio of 4 by 1. that means that the blue square has a height of 7/8 (because the top 1/4 * 1/2 = 1/8 is lopped off, and width (1/2). This means that the length of the line going from the center of the circle to the red point in the top left corner of the blue square is sqrt( (1/2)^2 + (7/8)^2 ) = sqrt( 1/4 + 49/64 ) = sqrt( 16/64 + 49/64 ) = sqrt( 65/64 ), which is approximately 1.008, or less than one percent removed from 1. So it's an excellent approximation, the error is probably smaller than the error you'll make in drawing the lines.