Geometry

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high importance~4 Q in Tier 137 formulas⚡ 19 shortcuts6 subtopics

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Every formula in this topic, grouped by subtopic. Print it and pin it above your desk.

Lines and angles

Angles on a line / around a point

180^\circ,\ 360^\circ

A straight line totals 180 degrees; one full turn totals 360.

Co-interior angles

a+b=180^\circ

The two inside angles on one side of a transversal, between parallel lines.

Complement / supplement

\text{supplement}-\text{complement}=90^\circ

Complement = 90 − x, supplement = 180 − x.

Equal pairs at parallels

\text{corresponding}=\text{alternate}=\text{vertically opposite}

Each of these pairs is equal.

Triangles and their centres

Angle sum and exterior angle

A+B+C=180^\circ,\quad \text{ext at }A=B+C

Exterior angle = sum of the two remote (far) interior angles.

Centroid division

AG:GD=2:1

G is the centroid on median AD; the vertex piece is twice the base piece.

Incentre angle

\angle BIC=90^\circ+\dfrac{A}{2}

I = incentre, where the angle bisectors meet.

Circumcentre angle

\angle BOC=2A

O = circumcentre; the angle at O stands on the same arc BC as angle A.

Orthocentre angle

\angle BHC=180^\circ-A

H = orthocentre, where the altitudes meet.

Apollonius (median length)

m_a^2=\dfrac{2b^2+2c^2-a^2}{4}

Median to side a of a triangle with sides a, b, c.

Isosceles median to base

m=\sqrt{a^2-\left(\dfrac{b}{2}\right)^2}

a = equal side, b = base.

Heron's area

K=\sqrt{s(s-a)(s-b)(s-c)},\ s=\dfrac{a+b+c}{2}

s = semi-perimeter (half the perimeter).

Congruence, similarity and BPT

Similarity ratios

\frac{a_1}{a_2}=k,\quad \frac{P_1}{P_2}=k,\quad \frac{K_1}{K_2}=k^2

a = side, P = perimeter, K = area; k = scale factor.

Areas from perimeters

\frac{K_1}{K_2}=\left(\frac{P_1}{P_2}\right)^2

Square a length ratio to get the area ratio; take a root to go back.

BPT (Thales)

DE\parallel BC\Rightarrow\frac{AD}{DB}=\frac{AE}{EC}

A line parallel to one side cuts the other two sides in the same ratio.

Midpoint theorem

D,E\ \text{midpoints}\Rightarrow DE=\frac{BC}{2}

The join of two midpoints is half the third side and parallel to it.

Angle bisector theorem

\frac{BD}{DC}=\frac{AB}{AC}

The bisector of angle A splits BC in the ratio of the sides AB and AC.

Pythagoras theorem and triplets

Pythagoras

h^2=p^2+b^2

h = hypotenuse (longest side, opposite the right angle).

Missing leg

p=\sqrt{h^2-b^2}

Subtract when a leg is missing.

Median to hypotenuse

m=\dfrac{h}{2}

The median drawn from the right-angle corner.

Rectangle diagonal

d=\sqrt{l^2+b^2}

The corner angles of a rectangle are right angles.

Triangle type test

a^2\lessgtr b^2+c^2

a = longest side; equal means right, greater means obtuse, smaller means acute.

Quadrilaterals and polygons

Quadrilateral angle sum

A+B+C+D=360^\circ

Any four-sided figure.

Parallelogram angles

A+B=180^\circ,\quad A=C

Adjacent angles supplement; opposite angles equal.

Cyclic quadrilateral

A+C=180^\circ,\quad B+D=180^\circ

Opposite corners on one circle.

Rhombus

K=\frac{1}{2}d_1d_2,\quad a=\sqrt{\left(\frac{d_1}{2}\right)^2+\left(\frac{d_2}{2}\right)^2}

d1, d2 = diagonals; they cross at right angles.

Trapezium

K=\frac{1}{2}(a+b)h

a and b are the two parallel sides; h is the gap between them.

Parallelogram

K=bh

Height is measured perpendicular to the base.

Regular polygon

\text{ext}=\frac{360^\circ}{n},\quad \text{int}=180^\circ-\text{ext},\quad \text{diagonals}=\frac{n(n-3)}{2}

n = number of sides.

Circles: chords, tangents, secants and cyclic angles

Chord from distance

\ell=2\sqrt{r^2-d^2}

d = distance from centre to chord.

Equal chords

\ell_1=\ell_2\Rightarrow d_1=d_2

Equal chords sit equally far from the centre.

Centre vs circumference angle

\angle BOC=2\angle BAC

Both angles stand on chord BC.

Tangent length

PT=\sqrt{d^2-r^2}

P is d from the centre of a circle of radius r.

Tangent-secant

PT^2=PA\cdot PB

Tangent squared = outside part times whole secant.

Intersecting chords

PA\cdot PB=PC\cdot PD

Two chords crossing inside the circle.

Alternate segment

\angle(\text{tangent},\ \text{chord})=\angle\text{ in alternate segment}

The angle between a tangent and a chord equals the angle the chord makes on the far side.

Common tangents (transverse / direct)

L_T=\sqrt{d^2-(r_1+r_2)^2},\quad L_D=\sqrt{d^2-(r_1-r_2)^2}

d = distance between centres.