The line AB is a diameter of the circle with center O. We pick a random point P on the circumference of the circle and then draw a line to P from A and a line to P from B. The angle APB looks like a right angle, look carefully, the location of P make no difference, this is Thales theorem. Formally, Thales's theorem states that if A,B, and P are distict points on a circle and the line AB is a diameter, the angle APB is a right angle.

The theorem has been atributted to Thales of Miletus who lived in ancient Greece. I am sure the ancient Greeks were very clever, but giving them the credit for most of ancient mathematics is not very credible, perhaps other civillisation also were aware of this theorem independently, if you have any insight on this then email me.

Can be used for all triangles, not just right angled triangles. The cosine rule formula is derived by dividing a scalene triangle into two right angled triangles.

The unit circle which is a circle with radius 1 is the fundamental concept behind trigonometry.

We select a random point P on the circumference of the circle.

A line is drawn from A to P and from B to P.

The lines are OA, OP, and OB are radii of the circle and are labelled as r.

Triangle OPB is an isosceles triangle ∠OPB= ∠OBP=x as shown.

Triangle OPA is an isosceles triangle ∠OPA= ∠OAP=y as shown.

The final step consider the triangle APB, the angles add up to 180 degrees: $$ x+(x+y)+y=180 $$ simplifying: $$ 2x+2y=180 $$ dividing all terms by 2: $$ x+y=90$$ ∠ therefore APB=90 degrees, and that is the proof Thale's theorem. The Pythagorus theorem can be applied to triangle ABP where the hypotenuse is the diameter.

Thales's theorem is probably the best known circle theorem, and there many more and some these will be discussed in future posts, watch this space.