Choose three of the given points.
Find the center and radius of the unique circle that passes through these three points.
Show that the distance from the circle center to the fourth point is equal to the radius of the circle. Alternatively, show that the fourth point satisfies the equation of the circle that can be written directly using the coordinates of the center and the square of the radius.
The perpendicular bisector of any chord passes through the center of the circle. Beginning with the chord
Since the slope of a line perpendicular to a line is the negative reciprocal of the slope of the line, the slope of the perpendicular bisector must be
And the midpoint coordinates:
So the midpoint of
From this information, we write the equation of the perpendicular bisector of chord
Similarly, an equation of the perpendicular bisector of chord
The details are an exercise for the student.
From these two equations, we can find the point of intersection of the two perpendicular bisectors.
Hence, the point of intersection, and therefore the center of the circle is
The distance from the center to any of the three points is the radius of the circle and can be found using the distance formula:
So if the distance from the center to the fourth point is equal to
Alternatively, we note that the equation of the circle is:
And since
Point D is on the circle. Q.E.D.
John
My calculator said it, I believe it, that settles it
