Circle Theorems

Define Tangent

A straight line that touches the circumference of a circle at one single point without cutting across it a 90 degrees.

Define Circumference

The total distance all the way around its outer boundary

Define Chord

A straight line segment whose end-points both lie on the circumference of a circle

Define Segment

The region of a circle enclosed by a chord and an arD

Define Radius

A straight line segment from the centre of a circle to any point ion its circumference.

Define Diameter

A straight line segment that passes through the center of a circle and has its endpoints on the circumference.

Rule 1 : ANGLE IN A SEMICIRCLE = 90°

A triangle drawn from the two ends of a diameter will always make an angle of 90° where it hits the edge of the circle, no matter where it hits.

A TANGENT AND RADIUS MEET AT 90°

A tangent is a line that just touches the edge of a curve. If a tangent and radius meet at the same point, then the angle they make is exactly 90°.

TWO RADII CREATE AN ISOSCELES TRIANGLE

These triangles are constructed using two radii. Therefore they are isosceles triangles.

ANGLES IN THE SAME SEGMENT ARE EQUAL

All triangles drawn from a chord will have the same angle where they touch the circle. Also the two angles on opposite sides of the chord add up to 180°

ANGLE AT THE CENTRE IS TWICE THE ANGLE AT THE circumference

Within the same segment, the angle made at the centre of a circle is exactly double the angle formed by the same chord at the edge of the circle.

OPPOSITE ANGLES OF A CYCLIC QUADRILATERAL ADDS TO 180°

A cyclic quadrilateral is a four sided shape with every corner touching the circle. Both pairs of opposite angles add up to 180°

Rule 7 : EQUALITY OF TANGENTS FROM A POINT

The two tangents drawn from an outside point always touch the circle at the same distance. This creates a reflective situation with two congruent right angled triangles.

Rule 8 : ANGLE IN THE ALTERNATE SEGMENT IS EQUAL

The angle between the tangent and a chord is equal to the opposite angle in the triangle in the other segment. This is the most difficult to remember!

Rule 9 : THE PERPENDICULAR CHORD BISECTOR IS A DIAMETER

No matter where you draw the chord, the line that cuts it exactly in half at 90° will go through the center of the circle – the diameter line.

Chord Properties

The line drawn from the centre of a circle to the mid-point of a chord is at right angles to the chord

Chord Properties (intersecting chords)

Two chords instructing inside the circle - When two chords of a circle intellect at an angle interior, the product of the lengths of the two segments on one chord equal the product of the lengths of the two segments on the other chord. AE x EB = CE x ED

Chord Properties (intersecting secants)

Secant-Secant Theorem / Power of a Point (outside the circle) : When two secants are drawn from the same point outside of a circle, multiply the outside part by the entire length of equal secant. The products will be equal. EAB and EDC are the secants. AB and CD are the chords. AE x BE = CE x DE