Circle Theorem

 

Circle Theorem 1

The angle at the center is twice the angle at the circumference.

Circle Theorem 2

The angle in a semi-cicle is 90°.

Circle Theorem 3

Angles in the same segment are equal.

Circle Theorem 4

Opposite angles in a cyclic quadrilateral add up to 180°.

Circle Theorem 5

The lengths of the two tangents from a point to a circle are equal.

Circle Theorem 6

The angle between a tangent and a radius in a circle is 90°.

Circle Theorem 7

Alternate segment theorem:

The angle (α) between the tangent and the chord at the point of contact (D) is equal to the angle (β) in the alternate segment*.

Circle Theorem 8

Perpendicular from the centre bisects the chord:


Circle Theorems for Arcs and Chords:

  • If two chords are congruent, then their corresponding arcs are congruent.
  • If the diameter or radius is perpendicular to a chord, then it bisects the chord and its arc.
  • In the same circle or congruent circles, two chords are congruent if and only if they are equidistant from the center.