Quick Revision

Lines and Angles

• Complementary angles add to $90^\circ$, supplementary to $180^\circ$; angles at a point to $360^\circ$.
• Vertically opposite angles are equal; a linear pair adds to $180^\circ$.
• Parallel lines and a transversal: corresponding and alternate angles are equal, co-interior angles add to $180^\circ$.
• Bend rule: $\angle BED = x + y$. If the bend points away, the three angles add to $360^\circ$.

• Bisectors of co-interior angles meet at $90^\circ$.
• Trap: use the angles on the side the bend points to.

Polygons

• Interior angles add to $(n - 2) \times 180^\circ$; exterior angles always add to $360^\circ$.
• Regular polygon: each exterior angle $= \frac{360^\circ}{n}$, each interior angle $= 180^\circ - \frac{360^\circ}{n}$.
• Sides from an angle: $n = \frac{360}{\text{exterior}}$. Interior : exterior $= p : q$ gives $n = \frac{2(p + q)}{q}$.
• Diagonals: $\frac{n(n - 3)}{2}$. A decagon has 35.
• Only the pentagon has as many diagonals as sides.
• Trap: if $360$ divided by the exterior angle is not a whole number, no such regular polygon exists.

Quadrilaterals

• Angle sum $360^\circ$. Bisectors of $\angle A$ and $\angle B$ meet at $\frac{\angle C + \angle D}{2}$.
• Parallelogram: adjacent angles add to $180^\circ$, diagonals bisect each other, $d_1^2 + d_2^2 = 2(a^2 + b^2)$.
• Rectangle: equal diagonals. Square: diagonal $a\sqrt{2}$.
• Rhombus: diagonals are perpendicular bisectors of each other, so $d_1^2 + d_2^2 = 4a^2$.
• Trapezium: mid-segment $= \frac{AB + CD}{2}$; diagonals cut each other in the ratio $AB : CD$.
• Kite: diagonals are perpendicular.
• Trap: halve the diagonals before using Pythagoras.

Triangle Angles and Sides

• Angle sum $180^\circ$; an exterior angle equals the sum of the two interior opposite angles.
• The larger side faces the larger angle.
• Triangle inequality: $|a - b| < c < a + b$, strictly.
• With $c$ the longest side: $a^2 + b^2 > c^2$ acute, $= c^2$ right, $< c^2$ obtuse.
• Isosceles: equal sides face equal angles. Equilateral: $60^\circ$ each, height $\frac{\sqrt{3}}{2}a$.
• [Tier 2] Median: $AB^2 + AC^2 = 2(AD^2 + BD^2)$.
• Trap: in isosceles chains, push one angle round using exterior angles.

Congruence

• Rules: SSS, SAS (the angle between the two sides), ASA, AAS, and RHS (right angle, hypotenuse, side).
• Not rules: SSA and AAA. Equal angles alone only give similar triangles.
• $\triangle ABC \cong \triangle PQR$ pairs A with P, B with Q, C with R, so $AB = PQ$ and $\angle B = \angle Q$.
• CPCT then gives every other matching part.
• The bisector from the top of an isosceles triangle makes two congruent halves, so it is also the median and altitude.
• Trap: equal area or perimeter does not mean congruent.

Pythagoras and Right Triangles

• Triplets: 3, 4, 5; 5, 12, 13; 8, 15, 17; 7, 24, 25; 20, 21, 29; 9, 40, 41; 12, 35, 37; 11, 60, 61; and their multiples.
• Odd leg $m$: the others are $\frac{m^2 - 1}{2}$ and $\frac{m^2 + 1}{2}$.
• 30-60-90 sides: $1 : \sqrt{3} : 2$, shortest side facing $30^\circ$.
• 45-45-90 sides: $1 : 1 : \sqrt{2}$.
• The median to the hypotenuse is half the hypotenuse.
• Trap: when the hypotenuse is given, subtract the squares.

Triangle Centres

• Incentre: $\angle BIC = 90^\circ + \frac{A}{2}$. Excentre: $\angle BEC = 90^\circ - \frac{A}{2}$.
• Circumcentre: $\angle BOC = 2A$, or $360^\circ - 2A$ if A is obtuse.
• Orthocentre: $\angle BHC = 180^\circ - A$, or $A$ if B or C is obtuse.
• Centroid: $AG = \frac{2}{3}AD$. Medians from B and C at $90^\circ$ give $AB^2 + AC^2 = 5BC^2$.
• Bisector and altitude from A: $\frac{|B - C|}{2}$ apart.
• [Tier 2] Euler line: $HG = 2GO$.
• Trap: swapping the incentre and excentre (common in the PYQ Practice Sets).

Inradius and Circumradius

• Any triangle: $r = \frac{\Delta}{s}$ and $R = \frac{abc}{4\Delta}$.
• Right triangle: $r = \frac{a + b - c}{2}$, $R = \frac{c}{2}$.
• Equilateral: $R = \frac{a}{\sqrt{3}}$, $r = \frac{a}{2\sqrt{3}}$, $R = 2r$, and $r : R : h = 1 : 2 : 3$.
• Incircle tangent lengths from A, B, C: $s - a$, $s - b$, $s - c$.
• [Tier 2] $OI^2 = R^2 - 2Rr$.
• Trap: the touch point D on BC gives $BD = s - b$, with $b = CA$.

Similar Triangles

• Tests: AA, SAS, SSS.
• Sides, perimeters, heights, medians and bisectors all follow the scale $k$; areas follow $k^2$.
• Altitude AD to the hypotenuse of a right triangle: $AD^2 = BD \times DC$, $AB^2 = BD \times BC$, $AC^2 = DC \times BC$, $AD = \frac{AB \times AC}{BC}$.
• $\angle ADC = \angle BAC$ with D on BC gives $AC^2 = DC \times BC$.
• Shadows and lamps: match the two similar triangles.
• Trap: from an area ratio, take the square root before finding a length.

Proportionality Theorems

• BPT: $DE \parallel BC$ gives $\frac{AD}{DB} = \frac{AE}{EC}$, and the converse holds.
• Scale $\frac{AD}{AB}$; area ADE : DBCE $= m^2 : (m + n)^2 - m^2$ for $AD : DB = m : n$.
• Midpoint theorem: the join of two midpoints is half the third side.
• Angle bisector: $\frac{BD}{DC} = \frac{AB}{AC}$. [Tier 2] The external bisector splits BC externally in the same ratio.
• Trapezium: $\frac{AO}{OC} = \frac{AB}{CD}$, and triangles AOD and BOC have equal areas.
• Trap: $\frac{DE}{BC} = \frac{AD}{AB}$, not $\frac{AD}{DB}$.

Chords and Arcs

• The perpendicular from the centre bisects a chord: $d^2 + \left(\frac{l}{2}\right)^2 = r^2$.
• Equal chords are equally far from the centre; the nearer chord is longer.
• Parallel chords: $d_1 - d_2$ apart on the same side, $d_1 + d_2$ on opposite sides.
• Common chord of two circles: perpendicular to the line of centres and bisected by it.
• Chords crossing inside: $AE \times EB = CE \times ED$.
• Trap: use half the chord.

Angles in a Circle

• The angle at the centre is twice the angle at the circumference on the same arc.
• Angles in the same segment are equal; the angle in a semicircle is $90^\circ$.
• Equal chords subtend equal angles.
• With C on the major arc: $\angle OAB = 90^\circ - \angle ACB$.
• [Tier 2] Chords crossing inside: angle $=$ half the sum of the arcs; secants meeting outside: half the difference.
• Trap: from the minor arc, the angle is $180^\circ - \frac{1}{2}\angle AOB$.

Cyclic Quadrilaterals

• Opposite angles add to $180^\circ$.
• An exterior angle equals the interior opposite angle.
• If one pair of opposite angles adds to $180^\circ$, the quadrilateral is cyclic.
• Rectangles, squares and isosceles trapeziums are cyclic; a parallelogram is cyclic only if it is a rectangle.
• AB and DC produced meet at P, AD and BC at Q, with A the corner away from both: $2\angle A + \angle P + \angle Q = 180^\circ$.
• Trap: all four corners must lie on the circle.

Tangents

• A tangent is perpendicular to the radius at the point of contact.
• Tangent length from P: $\sqrt{OP^2 - r^2}$.
• The two tangents are equal, and $\angle APB + \angle AOB = 180^\circ$.

• Alternate segment: tangent-chord angle $=$ angle in the other segment.
• Tangent and secant: $PT^2 = PA \times PB$.
• Quadrilateral round a circle: $AB + CD = AD + BC$.
• Trap: OP is from the centre, not from the circle.

Two Circles and Common Tangents

• Touching: $d = R + r$ externally, $d = R - r$ internally.
• Common tangents: 4 apart, 3 touching externally, 2 cutting, 1 touching internally, 0 one inside the other.
• Direct $\sqrt{d^2 - (R - r)^2}$; transverse $\sqrt{d^2 - (R + r)^2}$; touching externally, direct $= 2\sqrt{Rr}$.
• Three touching circles: each radius is half the perimeter of the centre triangle minus the opposite side.
• [Tier 2] Circle under two touching circles on a line: $\frac{1}{\sqrt{x}} = \frac{1}{\sqrt{R}} + \frac{1}{\sqrt{r}}$.
• Trap: direct goes with $R - r$.