IEB Grade 12 Euclidean Geometry Cheat Sheet

Euclidean geometry is about 40 marks of Paper 2. Every statement needs a reason, so this sheet gives the accepted short form for each.

Circle theorems (reason in brackets)

• Line from centre perpendicular to a chord bisects it (line from centre $\perp$ chord).

• Line from centre to midpoint of a chord is perpendicular to it (line from centre to midpt of chord).

• Angle at the centre is twice the angle at the circumference ($\angle$ at centre $=2\times \angle$ at circumference).

• Angles in the same segment are equal ($\angle$s in the same seg).

• Angle in a semicircle is 90° ($\angle$ in semi-circle).

• Equal chords subtend equal angles (equal chords; equal $\angle$s).

Cyclic quadrilaterals and tangents

• Opposite angles add to 180° (opp $\angle$s of cyclic quad).

• Exterior angle equals interior opposite angle (ext $\angle$ of cyclic quad).

• Prove cyclic with a converse: opp $\angle$s supplementary, ext $\angle$ = int opp $\angle$, or equal angles on the same side.

• Tangent is perpendicular to the radius (tan $\perp$ radius).

• Tangents from a common point are equal (tans from common pt).

• Tangent-chord angle equals the angle in the alternate segment (tan chord theorem).

Proportion and similarity

• Line parallel to one side divides the other two in proportion (line $\parallel$ one side of $△$; name the lines).

• Midpoint theorem: joins midpoints, parallel to and half the third side (midpt theorem).

• Equal heights: area ratio = base ratio.

• Equiangular triangles are similar (reason: equiangular $△$s); then sides are in proportion (reason: $|||$ $△$s).

• Altitude in a right triangle: $A{B}^2=AD\cdot AC$, $B{D}^2=AD\cdot DC$.

• 'Hence prove' a product: cross-multiply the ratio from the similar triangles.

Examinable proofs

• From Grade 11 (acute-angle case): line from centre to midpoint of chord is $\perp$ chord; angle at centre; opposite angles of a cyclic quad; tan-chord.

• From Grade 12: proportionality; equiangular triangles are similar; the perpendicular from the right angle to the hypotenuse gives similar triangles.

• Converses are not examinable as proofs. No concurrency and no proof by contradiction.

Know by heart

• All reasons in their accepted short form

• The seven examinable proofs with their constructions

On the formula sheet

• Nothing; all theorems and reasons must be known

Common traps

• Statements without reasons lose marks.

• Using cyclic quad properties before proving the quad is cyclic.

• Naming similar triangles in the wrong order.

• Assuming a line is a diameter or tangent because it looks like one.

Related topics

• circle geometry theorems
• cyclic quadrilaterals
• tangent chord theorem
• proportionality theorem
• similar triangles