Geometry
🔒 Log in to trackhigh importance~4 Q in Tier 136 formulas⚡ 19 shortcuts6 subtopics
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Lines and angles
Angles on a line / around a point
\text{on a line}=180^\circ,\quad \text{round a point}=360^\circ
Co-interior angles
(a+b)^\circ=180^\circ\ \text{for allied angles between parallels}
Complement / supplement
x+(90^\circ-x)=180^\circ\cdot\tfrac12,\quad \text{supplement}-\text{complement}=180^\circ-2\cdot\text{angle}
Parallel line pairs
\text{corresponding}=\text{alternate}=\text{vertically opposite (each pair equal)}
Triangles and their centres
Angle sum & exterior angle
A+B+C=180^\circ,\quad \text{ext}=B+C
Centroid division
AG:GD=2:1\ \text{on median }AD
Incentre angle
\angle BIC=90^\circ+\frac{A}{2}
Circumcentre angle
\angle BOC=2A\ \text{(minor arc BC)}
Orthocentre angle
\angle BHC=180^\circ-A
Apollonius (median length)
m_a^2=\frac{2b^2+2c^2-a^2}{4}
Isosceles median
m_{\text{base}}=\sqrt{a^2-\left(\frac{\text{base}}{2}\right)^2}
Heron's area
K=\sqrt{s(s-a)(s-b)(s-c)},\ s=\frac{a+b+c}{2}
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
Areas from perimeters
\frac{K_1}{K_2}=\left(\frac{P_1}{P_2}\right)^2
BPT
DE\parallel BC\Rightarrow\frac{AD}{DB}=\frac{AE}{EC}
Midpoint theorem
D,E\ \text{midpoints}\Rightarrow DE=\frac{BC}{2}
Angle bisector theorem
\frac{BD}{DC}=\frac{AB}{AC}\ \text{(internal bisector of }A\text{)}
Pythagoras theorem and triplets
Pythagoras
h^2=p^2+b^2
Median to hypotenuse
m_{\text{hyp}}=\frac{h}{2}
Rectangle diagonal
d=\sqrt{l^2+b^2}
Triangle type test
a\ \text{longest}:\ a^2\gtrless b^2+c^2\Rightarrow \text{obtuse}/\text{acute}
Quadrilaterals and polygons
Quadrilateral angle sum
A+B+C+D=360^\circ
Cyclic quadrilateral
A+C=180^\circ,\quad B+D=180^\circ
Rhombus
K=\frac12 d_1d_2,\quad a=\sqrt{\left(\frac{d_1}{2}\right)^2+\left(\frac{d_2}{2}\right)^2}
Trapezium
K=\frac12(a+b)h
Parallelogram
K=bh
Regular polygon
\text{int}=\frac{(n-2)180^\circ}{n},\quad \text{ext}=\frac{360^\circ}{n},\quad \text{diagonals}=\frac{n(n-3)}{2}
Brahmagupta
K=\sqrt{(s-a)(s-b)(s-c)(s-d)}
Circles: chords, tangents, secants and cyclic angles
Chord from distance
\ell=2\sqrt{r^2-d^2}
Equal chords
\ell_1=\ell_2\Rightarrow d_1=d_2
Tangent–secant
PT^2=PA\cdot PB
Intersecting chords
PA\cdot PB=PC\cdot PD
Cyclic quadrilateral
A+C=180^\circ,\ \text{ext}=\text{interior opposite}
Alternate segment
\angle\text{tangent-chord}=\angle\text{in alternate segment}
Common tangents (transverse)
L_T=\sqrt{d^2-(r_1+r_2)^2}
Common tangents (direct)
L_D=\sqrt{d^2-(r_1-r_2)^2}