Geometric Proofs

Theorem

A statement that can be proved by a series of logical steps.

Axiom

Is a statement that is accepted to be true without proof.

Circle theorem 1

The angle at the centre of a circle is twice the angle at the circumference that subtends the same arc.

Circle theorem 2

Angles at the circumference of a circle subtended by the same arc are equal.

Circle theorem 3

A angle that is subtended at the circumference from the diameter is a right angle.

Circle theorem 4

Equal chords of a circle subtend equal angles at the centre. Conversely, equal angles at the centre subtend equal chords.

Circle theorem 5

A radius that bisects a chord is perpendicular to the chord.

Circle theorem 6

When two chords of a circle intersect, the product of the lengths of the intervals on one chord equals the product of the lengths of the interval on the other chord.

Circle theorem 7

A tangent drawn to a circle is perpendicular to the radius at the point of contact.

Circle theorem 8

Tangents are drawn to a circle from an external point are equal in length.

Circle theorem 9

The angle between the tangent and a chord is equal to the angle in the alternate segment.

Circle theorem 10

When a secant and a tangent are drawn from a external point , the square of the length of the tangent equals the product of the lengths to the circle on the secant.

Theorem 11

The opposite angles of a cyclic quadrilateral are supplementary. In the diagram a+b=180.

Theorem 12

The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.