Grade 12 Euclidean Geometry: Circle Theorems and Accepted Reasons

The circle theorems you must know

• The line from the centre to the midpoint of a chord is perpendicular to the chord (line from centre to midpt of chord).
• The angle at the centre is twice the angle at the circumference on the same arc (∠ at centre = 2 × ∠ at circumference).
• The angle in a semi-circle is 90° (∠s in semi-circle).
• Angles in the same segment are equal (∠s in same seg).
• Opposite angles of a cyclic quadrilateral add to 180° (opp ∠s cyclic quad).
• The exterior angle of a cyclic quadrilateral equals the interior opposite angle (ext ∠ cyclic quad).
• A tangent is perpendicular to the radius at the point of contact (tan ⊥ radius).
• Two tangents from the same point are equal (tans from common pt).
• The tangent–chord angle equals the angle in the alternate segment (tan-chord).

Proving a quadrilateral is cyclic

• Opposite angles add to 180°.
• The exterior angle equals the interior opposite angle.
• Two angles on the same side of a line subtended by the same segment are equal (converse of ∠s in same seg).

Worked proof

In a circle with centre O, AB is a diameter and C lies on the circle. Prove that ∠ACB = 90°. Join OC. OA = OB = OC (radii), so triangle OAC and triangle OBC are isosceles. Let ∠OAC = x and ∠OBC = y, so ∠OCA = x and ∠OCB = y (∠s opp equal sides).

The angles of triangle ABC add to 180°, so x + y + (x + y) = 180°, giving x + y = 90°. Since ∠ACB = x + y, ∠ACB = 90°. In the exam you may simply quote '∠s in semi-circle' unless the question says 'prove'.

Similar triangles and proportion

For the proportion theorem, a line parallel to one side of a triangle divides the other two sides proportionally (line ∥ one side of Δ). To prove triangles similar, show two pairs of equal angles (∠∠∠), then write the sides in matching order before forming ratios.

Write corresponding vertices in the same order, for example ΔABC ||| ΔDEF. Marks are lost when the ratio pairs the wrong sides.