← Topic 4 · Geometry
AMC 10/12 · Cheatsheet

Topic 4 · Geometry

Chapter 1 · Circle & area techniques

📋 Reference · always available
Power of a point (chords)
PA⋅PB=PC⋅PDPA\cdot PB = PC\cdot PD
Power of a point (tangent)
PT2=PA⋅PBPT^2 = PA\cdot PB
Shoelace (triangle)
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
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
∠central=2 ∠inscribed\angle_{\text{central}} = 2\,\angle_{\text{inscribed}}
Same arc, vertex on the circle.
Semicircle (Thales)
Inscribed angle on a diameter =90∘= 90^\circ.
Cyclic quadrilateral
∠A+∠C=180∘\angle A + \angle C = 180^\circ
Converse: opposite angles sum to 180∘180^\circ ⇒ cyclic.
Tangent–chord (alt segment)
Angle between tangent and chord = inscribed angle in the alternate segment.
Ptolemy
AC⋅BD=AB⋅CD+AD⋅BCAC\cdot BD = AB\cdot CD + AD\cdot BC
For cyclic ABCDABCD.