The general equation of a circle is:
(x - a) 2 - (y - b) 2 = r 2
Where (a, b) denotes the coordinate of the center, and r is the radius. See e.g.
For the circle in the problem, defined by:
(x - 132) 2 - (y - 132) 2 = 24, 649
The center is at: (132, 132) and the radius is:
Ö 24, 649 = 157
Then: AO = 157
But the horizontal distance from A to O (see reconstructed diagram) is 132.
By the Pythagorean theorem:
(Vertical distance from A to O)2 + (132) 2 = (157) 2 = 7, 225
Then: Vertical distance from A to O = Ö 7, 225 = 85
Now, <ACM and < ACN are inscribed angles with both = p/4 radians
The inscribed angle theorem states:
An inscribed angle is equal to one half the central angle that intercepts the same arc.
Therefore: <ACM = ½ < AOM
And: <ACN = ½ < AOM + < AOM
= <AON = p/2 radians
NM is a diameter (in the reconstructed circle) since <AON and <AOM are on opposite sides of AO and < AOM + < AON = p radians
NM is perpendicular to AO, since < AOM and < AON are p/2 radians relative to AO.
Then: The slope of AC = 85/ 132
The slope of NM = 132/ 85
M coordinates are: (132 + 85, 132 + 132) = (217, 264)
N coordinates are: (132 - 85, 132 - 132) = (47, 0)
