Grade 11 Euclidean Geometry

Core Circle Theorems

This covers every single acceptable reason for circles listed in the DBE 2015 Exam Guidelines. Forward theorems, converses, and shortcuts are all here.

1. Centre and Chords

line from centre ⊥ to chord

line from centre to midpt of chord

perp bisector of chord

The Bisector Rules

Forward (Examinable Proof): A line drawn from the centre perpendicular to a chord bisects the chord.

Converse 1: If a line from the centre bisects a chord, it must be perpendicular (90°) to the chord.

Converse 2: A perpendicular bisector of a chord ALWAYS passes through the centre of the circle.

→ View the formal proof for this theorem

2. Angle at Centre

∠ at centre = 2 × ∠ at circumference

The Arrowhead

Forward (Examinable Proof): The angle subtended by an arc at the centre of a circle is double the size of the angle subtended by the same arc at the circumference.

→ View the formal proof for this theorem

3. Angle in a Semi-Circle

∠s in semi circle OR diameter subtends right angle

chord subtends 90° OR converse ∠s in semi circle

The Diameter Rule

Forward: A diameter subtends an angle of exactly 90° at the circumference.

Converse: If an angle at the circumference is exactly 90°, the chord subtending it MUST be a diameter.

4. Angles in the Same Segment

∠s in the same seg

line subtends equal ∠s OR converse ∠s in the same seg

The "Bowtie" Concept

Forward: If two angles on the circumference are subtended by the same arc, they are equal.

Converse (Proving Cyclic Quads): If a line segment joining two points subtends equal angles at two other points on the same side, those four points are concyclic (they lie on a circle).

5. Equal Chords, Equal Angles

equal chords; equal ∠s

equal circles; equal chords; equal ∠s

The Mirror Concept

If two chords have the exact same length, they subtend the exact same size angles at the circumference. This is also true if the chords are in two separate circles that have the exact same radius (equal circles).

6. Cyclic Quadrilaterals

opp ∠s of cyclic quad

opp ∠s quad supp OR converse opp ∠s of cyclic quad

ext ∠ of cyclic quad

ext ∠ = int opp ∠ OR converse ext ∠ of cyclic quad

Opposite Angles and Exterior Angles

Forward (Examinable Proof): The opposite angles of a cyclic quadrilateral are supplementary (they add up to 180°).

Forward Shortcut: The exterior angle of a cyclic quad is exactly equal to the interior opposite angle.

Converses (Proving Cyclic Quads): You can prove a shape is a cyclic quad by proving the exterior angle equals the interior opposite, OR by proving opposite interior angles add to 180°.

→ View the formal proof for this theorem

7. Tangents & Tan-Chord

tan ⊥ radius OR tan ⊥ diameter

line ⊥ radius OR converse tan ⊥ radius

Tans from common pt OR Tans from same pt

tan chord theorem

converse tan chord theorem OR ∠ between line and chord

The 90-Degree Collision & Alternate Segments

Radius: A tangent meets a radius at exactly 90°. (Converse: If a line meets a radius at 90°, it is officially a tangent).

External Point: Two tangents drawn from the same external point are equal in length.

Tan-Chord (Examinable Proof): The angle between a tangent to a circle and a chord drawn from the point of contact is equal to the angle in the alternate segment.

Tan-Chord Converse: If a line is drawn through the end-point of a chord, making an angle equal to the angle in the alternate segment, then that line is a tangent to the circle.

→ View the formal proof for the Tan-Chord theorem