Finding the center and radius of a circle (Guo Jin, Luke, Jerome)

1) How to find the center of a circle:

Assume we have a circle

We will use the property: angles in a semicircle = 90º.
This property shows that if a triangle is drawn inside a semicircle, the angle opposite the diameter will be 90º.
With this property in mind, let us draw a 90º angle at a point on the circumference of the circle.

The 2 lines from the 90º angle form 2 sides of a triangle inscribed inside a semicircle. This can be derived by working backwards from the property of angles in a semicircle. Since the triangle is inscribed inside the semicircle, the 3rd side of the triangle is the diameter of the semicircle, and thus the diameter of the circle itself.

We can repeat the process again, and draw another right angle. By extending the lines formed by the right angle, another triangle inscribed in the semicircle is formed. Again, the 3rd side of this triangle is the diameter.

The point where the two diameters intersect is the center of the circle. (All diameters pass through the center, and all diameters intersect only at the center.)


2) How to find the radius of a circle 

Method 1
Assuming we have another circle

Draw 2 lines of the same length joined at a 90º angle at the circumference. Join the lines with another line.

These lines form the 2 sides of an inscribed triangle in the semicircle (the semicircle is the circle divided in half). The third side of the triangle would be the diameter of the semicircle which the triangle is inscribed in. The semicircle's diameter is the diameter of the circle itself.

Draw another line down from the point where the lines meet. Since the lines are of equal length, the line drawn here will bisect the diameter, forming 2 radii. The line that is drawn down to meet the diameter is also another radius.

Therefore, the radius can be determined.

Other Methods:
If the center has already been found, using methods such as in the previous part,

Draw a line from the center to the circumference, on one side. This line is the radius.
This method can be used for any other circle which the center is already known.

Otherwise, calculation can be used. If the diameter, circumference or area of the circle is known, their respective formulas can be applied to obtain the radius.

Measurement can also be used.