Geometry Notes

Last reviewed 22 Sept 2026 · 6 min read

Lines and angles

Angles on a straight line add to $180°$ ; angles round a point, to $360°$ .
Vertically opposite angles are equal.
With a transversal across parallel lines: corresponding angles equal, alternate angles equal, co-interior (allied) angles supplementary.

Triangles

FormulaBasic results

Angle sum $= 180°$ ; an exterior angle equals the sum of the two opposite interior angles.
Triangle inequality: the sum of any two sides exceeds the third; the greater side faces the greater angle.
Pythagoras (right angle at C): $a^{2} + b^{2} = c^{2}$ . Common triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25, 9-40-41, and their multiples.
Equilateral side $a$ : height $\frac{3}{2} a$ , area $\frac{3}{4} a^{2}$ .

Congruence and similarity

Congruence: SSS, SAS, ASA, AAS, RHS — the triangles are identical.
Similarity: AA, SSS (proportional), SAS (proportional) — the same shape, scaled.
In similar triangles, corresponding sides are in a fixed ratio $k$ ; areas are in the ratio $k^{2}$ .

FormulaTwo theorems that appear constantly

Basic proportionality (Thales): a line parallel to one side of a triangle divides the other two sides in the same ratio.
Midpoint theorem: the segment joining the midpoints of two sides is parallel to the third and half its length.

The four centres

Centre	Meeting point of	Property
Centroid	the three medians	divides each median $2 : 1$ from the vertex
Incentre	the three angle bisectors	centre of the inscribed circle
Circumcentre	the perpendicular bisectors of the sides	centre of the circumscribed circle
Orthocentre	the three altitudes	—

In a right-angled triangle the circumcentre is the midpoint of the hypotenuse, so the median to the hypotenuse is half of it. In an equilateral triangle all four coincide.

Quadrilaterals

Parallelogram: opposite sides and angles equal; diagonals bisect each other.
Rectangle: a parallelogram with equal diagonals.
Rhombus: a parallelogram whose diagonals bisect each other at right angles; side $= (d_{1} /2)^{2} + (d_{2} /2)^{2}$ .
Square: both of the above; diagonal $= a 2$ .
Trapezium: one pair of parallel sides; the line joining the midpoints of the non-parallel sides is half their sum.

Polygons

FormulaAngles and diagonals of an n-gon

Sum of interior angles = (n - 2) \times 180°, Sum of exterior angles = 360°

For a regular polygon: each interior angle $= \frac{( n - 2 ) 180°}{n}$ , each exterior angle $= \frac{360°}{n}$ . Number of diagonals $= \frac{n ( n - 3 )}{2}$ .

Circles

FormulaChords, tangents and angles

The perpendicular from the centre to a chord bisects it: $r^{2} = d^{2} + (ℓ /2)^{2}$ for a chord of length $ℓ$ at distance $d$ .
Equal chords are equidistant from the centre.
A tangent is perpendicular to the radius at the point of contact; the two tangents from an external point are equal.
The angle at the centre is twice the angle at the circumference on the same arc; the angle in a semicircle is $90°$ .
Angles in the same segment are equal.
In a cyclic quadrilateral opposite angles add to $180°$ , and an exterior angle equals the opposite interior angle.
Power of a point $P$ outside the circle: $P A \times P B = P C \times P D = P T^{2}$ , where $P T$ is the tangent length.

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Geometry