Relation between triangle side lengths and one of its angles: c2=a2+b2-2abcosγ.
The inscribed angle theorem states that the angle formed by two points on a circle and an opposing point on the circle is half that formed by those first two points and the circle centre.
In a polygon with n sides, the sum of the interior angles is 180*(n-2).
Ancient theorem stating that the products of the line segments, created on two intersecting chords in a circle, are equal. More details here.
In a triangle we have sin(α)/a=sin(β)/b=sin(γ)/c=d, where d is the diameter of the circumcircle.
The line segment in a triangle joining the midpoint of two sides of the triangle is parallel to its third side and is also half the length of this side.
If a line reflects from a wall, meaning the angle of incidence equals the angle of reflection, one can mirror the situation in the wall making the the line a straight line.
Take a line through a given point P cutting a circle with centre M and radius r. The product of the distance of the point to one intersection point and the distance to the other intersection point is constant, which is called the power of the point. This power is also equal to |PM|2-r2.
If the point lies inside the circle, this theorem is equivalent to the Intersecting chords theorem. It is also the basis for the Tangent-secant theorem.
Ptolemy’s theorem is a relation between the sides and diagonals of a cyclic quadrilateral: S1*S3+S2*S4=D1*D2.
Ancient theorem relating side lengths in a right triangle: a2+b2=c2.