Download RZC-Factsheet-Geometry (Revision)

Geometry Cheatsheet
Definitions
Centres of triangles:
Incentre – Intersection of angle bisectors
(the incentre would be the centre of a circle
inscribed inside the triangle)
Circumcentre – intersection of
perpendicular bisectors (the circumcentre
would be the centre of the circle
circumscribing the triangle)
Centroid – Intersection of medians
Orthocentre – intersection of altitudes (i.e.
a line from a vertex to the opposite side
such that the altitude and this side are
perpendicular)
Generic Tips
1. Reflect on the significance of each piece of information given to you:
a. We have a tangent?
We’ll likely be able to use the Alternate Segment Theorem (which you should expect
to use a lot!). If there’s a chord attached, use it immediately. If there’s isn’t a chord,
consider adding an appropriate one so we can use the theorem!
The additional presence of the radius (or adding the radius) gives us a 90° angle.
b. Two circles touch?
We have a tangent. The centres of the circles and the point of contact are collinear,
and we can use the tips in (a).
c. We’re given the diameter?
The angle subtended by any point on the circumference is 90°.
2. Use variables to represent appropriate unknown angles/lengths. Form equations (e.g. using
Pythagoras, or comparing lengths) to find the values of these variables.
3. Look out for similar triangles whenever you notice angles that are the same. This allows us
to compare lengths.
4. Justify your assumptions. It’s incredibly easy to lose easy marks in the BMO this way.
5. Extend or add lines where necessary.
6. Remember that ‘if and only if’ requires us to prove in both directions.
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Recipes for angles
The reverse also applies for the first and last of these. i.e. If a triangle is right-angled, then the
hypotenuse is the diameter of the circumscribed circle. And if the opposite angles of a quadrilateral
add up to 180°, it’s a cyclic quadrilateral.
Exterior angle of triangle = sum of two other internal angles.
Recipes for lengths

Ptolemy’s Theorem: 𝑨𝑪 × 𝑩𝑫 = (𝑨𝑩 × 𝑫𝑪) + (𝑨𝑫 × 𝑩𝑪)
(i.e. product of the two diagonals of a cyclic quadrilateral is the sum of the
products of the opposite pairs of sides)

Intersecting Chord Theorem: 
𝑎𝑏 = 𝑥𝑦

Similar Triangles: Actively look out for these. One common
occurrence is the following on the right where one triangle is
embedded in the other.
If ∠𝐴𝐵𝐷 = ∠𝐶𝐴𝐷, triangles 𝐴𝐵𝐷 and 𝐴𝐶𝐷 are similar.
Angle Bisector Theorem
𝐵𝐷 𝐴𝐵
=
𝐷𝐶 𝐴𝐶

Intersecting Secant Length Theorem
𝑃𝐴 × 𝑃𝐵 = 𝑃𝐶 × 𝑃𝐷
 Radii of Circle: Add radii in to points of interest. Obviously, all radii will be the same length.
 Pythagoras! You can often add lines to create right-angled triangles so that you can
subsequently exploit Pythagoras Theorem. If ever in doubt, form a right-angled triangle. 
It’s worthwhile memorising this table: (don’t faff about drawing ‘magic triangles’ each time to work
these out – you should know these off-hand)
sin
45°
1
cos
√2
1
tan
√2
1
30°
1
2
√3
2
1
√3
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60°
√3
2
1
2
√3