← Topic 4 · Geometry
AMC 10/12 · Topic Cheatsheet

Topic 4 · Geometry

40 key results accumulated across 4 chapters.

Power of a point (chords)
Ch 1
PA⋅PB=PC⋅PDPA\cdot PB = PC\cdot PD
Power of a point (tangent)
Ch 1
PT2=PA⋅PBPT^2 = PA\cdot PB
Shoelace (triangle)
Ch 1
A=12∣x1(y2−y3)+x2(y3−y1)+x3(y1−y2)∣A=\tfrac12\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right|
Heron
Ch 1
A=s(s−a)(s−b)(s−c),  s=a+b+c2A=\sqrt{s(s-a)(s-b)(s-c)},\; s=\tfrac{a+b+c}{2}
Central / inscribed
Ch 1
∠central=2 ∠inscribed\angle_{\text{central}} = 2\,\angle_{\text{inscribed}}
Same arc, vertex on the circle.
Semicircle (Thales)
Ch 1
Inscribed angle on a diameter =90∘= 90^\circ.
Cyclic quadrilateral
Ch 1
∠A+∠C=180∘\angle A + \angle C = 180^\circ
Converse: opposite angles sum to 180∘180^\circ ⇒ cyclic.
Tangent–chord (alt segment)
Ch 1
Angle between tangent and chord = inscribed angle in the alternate segment.
Ptolemy
Ch 1
AC⋅BD=AB⋅CD+AD⋅BCAC\cdot BD = AB\cdot CD + AD\cdot BC
For cyclic ABCDABCD.
Triangle / polygon
Ch 2
△=180∘,    n-gon=(n−2)180∘\triangle=180^\circ,\;\; n\text{-gon}=(n-2)180^\circ
Inscribed angle
Ch 2
Half the central angle on the same arc; semicircle ⇒ 90∘90^\circ.
Similarity ratios
Ch 2
length:k,  area:k2,  volume:k3\text{length}:k,\;\text{area}:k^2,\;\text{volume}:k^3
Centroid
Ch 2
G=A+B+C3G=\tfrac{A+B+C}{3}
Divides each median 2:12{:}1 from the vertex.
Incenter / circumcenter
Ch 2
Angle bisectors ⇒ II (incircle); ⊥\perp bisectors ⇒ OO (circumcircle).
Orthocenter
Ch 2
Altitudes meet at HH; for right triangle, HH = right-angle vertex.
Euler line
Ch 2
OG→:GH→=1:2\overrightarrow{OG}:\overrightarrow{GH} = 1{:}2
OO, GG, HH collinear (non-equilateral).
Stewart's theorem
Ch 2
b2m+c2n−ad2=amnb^2 m + c^2 n - a d^2 = a m n
Median length
Ch 2
ma=122b2+2c2−a2m_a = \tfrac12 \sqrt{2b^2 + 2c^2 - a^2}
Angle-bisector length
Ch 2
ta=2b+cbc⋅s(s−a)t_a = \tfrac{2}{b+c}\sqrt{bc \cdot s(s-a)}
Sum-to-product
Ch 2
sin⁡A+sin⁡B=2sin⁡A+B2cos⁡A−B2\sin A+\sin B = 2\sin\tfrac{A+B}{2}\cos\tfrac{A-B}{2}
Roots-of-unity sum
Ch 2
∑k=0n−1cos⁡ ⁣(θ+2πkn)=0  (n≥2)\sum_{k=0}^{n-1}\cos\!\left(\theta+\tfrac{2\pi k}{n}\right)=0\;(n\ge2)
$\sin A = \sin B$
Ch 2
A=B+2πkA = B + 2\pi k or A=π−B+2πkA = \pi - B + 2\pi k.
SOH-CAH-TOA
Ch 2
sin⁡=oh,cos⁡=ah,tan⁡=oa\sin=\tfrac{o}{h},\cos=\tfrac{a}{h},\tan=\tfrac{o}{a}
Law of cosines / sines
Ch 2
c2=a2+b2−2abcos⁡C,    asin⁡A=bsin⁡Bc^2=a^2+b^2-2ab\cos C,\;\; \tfrac{a}{\sin A}=\tfrac{b}{\sin B}
Pythagorean identity
Ch 2
cos⁡2θ+sin⁡2θ=1\cos^2\theta+\sin^2\theta=1
Distance
Ch 3
d=(Δx)2+(Δy)2d=\sqrt{(\Delta x)^2+(\Delta y)^2}
Circle
Ch 3
(x−h)2+(y−k)2=r2(x-h)^2+(y-k)^2=r^2
Perp. slopes
Ch 3
m1m2=−1m_1 m_2 = -1
Transforms
Ch 3
90∘ ccw:(x,y)→(−y,x)90^\circ\text{ ccw}: (x,y)\to(-y,x)
Volumes
Ch 4
Bh,    13Bh,    43πr3Bh,\;\; \tfrac13 Bh,\;\; \tfrac43\pi r^3
Sphere SA
Ch 4
4πr24\pi r^2
Point–line distance
Ch 4
d=∣ax0+by0+c∣a2+b2d=\tfrac{|ax_0+by_0+c|}{\sqrt{a^2+b^2}}
Line–circle
Ch 4
Substitute → quadratic; discriminant gives 0/1/2 intersections.
Sphere in cone (inradius)
Ch 4
r=RhR+R2+h2r = \tfrac{Rh}{R + \sqrt{R^2+h^2}}
Sphere : cylinder volume
Ch 4
VS:VC=2:3V_S : V_C = 2:3
Frustum volume
Ch 4
V=πh3(R2+Rr+r2)V = \tfrac{\pi h}{3}(R^2 + Rr + r^2)
Parabola
Ch 4
y=x24p:focus (0,p), directrix y=−py = \tfrac{x^2}{4p}: \text{focus }(0,p),\text{ directrix }y{=}{-}p
Ellipse
Ch 4
x2a2+y2b2=1,  c2=a2−b2\tfrac{x^2}{a^2}+\tfrac{y^2}{b^2}=1,\;c^2 = a^2 - b^2
Hyperbola
Ch 4
x2a2−y2b2=1,  c2=a2+b2\tfrac{x^2}{a^2}-\tfrac{y^2}{b^2}=1,\;c^2 = a^2 + b^2
Eccentricity
Ch 4
e=c/a:  parabola =1,  ellipse <1,  hyperbola >1e = c/a:\;\text{parabola }=1,\;\text{ellipse }<1,\;\text{hyperbola }>1