Proving the sine rule
Let ABC be a triangle, and E be its circumcircle. Let D be a point on E such that COD is a diameter.
Then angles CDB and CAB have the same measure, and CBD is a right angle; the diameter of the circle is a/sin(A). Using similar arguments, the diameter of the circle is b/sin(B) and c/sin(C), so a/sin(A) = b/sin(B) = c/sin(C).