Alternate segment theorem

For any circle, the angle formed between the tangent and the chord through the point of contact of the tangent is equal to the angle formed by the chord in the alternate segment.

Cosine rule

Relation between triangle side lengths and one of its angles: c2=a2+b2-2abcosγ.

Inscribed angle theorem

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.

Interior angle sum

In a polygon with n sides, the sum of the interior angles is 180*(n-2).

Intersecting chords theorem

Ancient theorem stating that the products of the line segments, created on two intersecting chords in a circle, are equal. More details here.

Law of sines

In a triangle we have sin(α)/a=sin(β)/b=sin(γ)/c=d, where d is the diameter of the circumcircle.

Midpoint theorem

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.

Mirror trick

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.

Power-of-a-point theorem

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.