AMC 10/12 · Cheatsheet

Topic 4 · Geometry

Chapter 2 · Angles, similarity & trigonometry

📋 Reference · always available

Triangle / polygon

\triangle=180^\circ,\;\; n\text{-gon}=(n-2)180^\circ

Inscribed angle

Half the central angle on the same arc; semicircle ⇒

90^\circ

Similarity ratios

\text{length}:k,\;\text{area}:k^2,\;\text{volume}:k^3

Centroid

G=\tfrac{A+B+C}{3}

Divides each median

2{:}1

from the vertex.

Incenter / circumcenter

Angle bisectors ⇒

I

(incircle);

\perp

bisectors ⇒

O

(circumcircle).

Orthocenter

Altitudes meet at

H

; for right triangle,

H

= right-angle vertex.

Euler line

\overrightarrow{OG}:\overrightarrow{GH} = 1{:}2

O

G

H

collinear (non-equilateral).

Stewart's theorem

b^2 m + c^2 n - a d^2 = a m n

Median length

m_a = \tfrac12 \sqrt{2b^2 + 2c^2 - a^2}

Angle-bisector length

t_a = \tfrac{2}{b+c}\sqrt{bc \cdot s(s-a)}

Sum-to-product

\sin A+\sin B = 2\sin\tfrac{A+B}{2}\cos\tfrac{A-B}{2}

Roots-of-unity sum

\sum_{k=0}^{n-1}\cos\!\left(\theta+\tfrac{2\pi k}{n}\right)=0\;(n\ge2)

$\sin A = \sin B$

A = B + 2\pi k

A = \pi - B + 2\pi k

SOH-CAH-TOA

\sin=\tfrac{o}{h},\cos=\tfrac{a}{h},\tan=\tfrac{o}{a}

Law of cosines / sines

c^2=a^2+b^2-2ab\cos C,\;\; \tfrac{a}{\sin A}=\tfrac{b}{\sin B}

Pythagorean identity

\cos^2\theta+\sin^2\theta=1