A diameter and a point on the circle
A diameter is a straight line across a circle that passes through its center. Call its two ends A and B. Now choose any other point on the circle, call it P, and join it to A and to B. This makes a triangle APB, with the diameter as one of its sides.
The diameter AB, and lines from its two ends to a point P on the circle.
The angle at P is a right angle
The angle at P, between PA and PB, is 90°. Move P anywhere else on the circle and the angle at P is still 90°. This is the angle in a semicircle: the angle that a diameter makes at any point on the circle is a right angle. It is named that way because P lies on one of the two halves of the circle that the diameter cuts off, the semicircles.
Drag P along the circle. The angle measured at P reads 90° wherever P goes.
Why: two isosceles triangles
Call the center O and draw the radius OP. OA, OB and OP are all radii of the same circle, so they are all equal. The radius OP splits triangle APB into two triangles, AOP and BOP, and each of them has two equal sides, so each is isosceles.
In an isosceles triangle, the two angles opposite the equal sides are equal. In triangle AOP, OA = OP, so the angle at A is equal to the angle OPA. Call each of them a. In triangle BOP, OB = OP, so the angle at B is equal to the angle OPB. Call each of them b.
Now add up the angles of the whole triangle APB. The angle at A is a, the angle at B is b, and the angle at P is made of both parts, a + b. The angles of a triangle add to 180°, so a + b + a + b = 180°. That is 2(a + b) = 180°, so a + b = 90°, and a + b is the angle at P.
Nothing in this argument depended on where P was, so it holds for every point on the circle. In the figure below, P is placed so that a = 55° and b = 35°, and 55° + 35° = 90°.
The radius from the center O to P cuts triangle APB into two isosceles triangles. The two angles marked a are equal, and so are the two angles marked b.
Using the right angle
Because the angle at P is 90°, the other two angles of triangle APB add to 90°. If the angle at A is 28°, the angle at B is 180° − 90° − 28° = 62°.
The diameter is the hypotenuse of the right-angled triangle, so Pythagoras' theorem applies to it. If PA = 6 cm and PB = 8 cm, the diameter is cm, and the radius of the circle is 5 cm.
The other way round: finding the center
The theorem also works backwards: if P is on the circle and the angle APB is 90°, with A and B on the circle too, then AB is a diameter. To see why, draw the diameter that starts at A, and call its other end D. By the theorem, the angle APD is 90° as well. So PB and PD both leave P at a right angle to PA. There is only one line through P at a right angle to PA, and it meets the circle at only one point besides P. So D is B, and AB is a diameter.
This finds the center of any circle, such as a round plate, with a set square, a flat triangle with one right-angled corner. Put the right-angled corner on the edge of the circle, and mark the two points where the edges of the set square cross the circle. Those two points are the ends of a diameter: join them. Do the same with the corner at another point of the circle to draw a second diameter. The center is where the two diameters cross.
Two right angles with their corners on the circle. The arms of each meet the circle at the two ends of a diameter, and the two diameters cross at the center.
The usual mistakes
Putting the right angle at the wrong corner. The 90° is at P, the point on the circle across from the diameter, not at A or B, the ends of the diameter.
Forgetting the right angle when finding the third angle. If the angle at A is 28°, then 180° − 28° = 152° leaves out the 90° at P, and 90° + 28° = 118° adds it instead of taking it away. The angle at B is 180° − 90° − 28° = 62°.
Using the theorem when AB is not a diameter. If the line from A to B does not pass through the center, the angle at P is not 90°.