Afly's notes

About Afly

Friends with Yrock and ConfidentFlamingo

User # 1013218 Profile

Competitions I have participated in:

AMC8 * 1

AMC10A * 1

AMC10B * 1

MathLeague Qualifications * 7

MathLeague District * 2

MathLeague District Countdown * 1

MathLeague State * 1

MathLeague State Countdown * 1

MathLeague National * 1

MathLeague National Countdown * 1

MathLeague International * 1

Some of my asymptote creations

These are a random collection of diagrams and text that I have made. YOU ARE ALLOWED TO USE THESE IMAGES AND/OR TEXT IN YOUR PAGES OR USE THEM FOR REFERENCE.

I try to put them in the order of how much technique & technical points there are in them

Enough talking. Let's get into the pictures.

Triangle

$[asy] pair A=(0,0); pair B=(2,0); pair C=(1,sqrt(3)); draw(A--B--C); draw(A--C--B); [/asy]$

$\text{That is a equilateral triangle with side length 2!}$

$\text{The height is } \sqrt{2^2-1^2} = \sqrt{3}$

$\text{The area is } 2 \cdot \sqrt{3} \cdot \frac{1}{2} = \boxed{\sqrt{3}}$

$\dots$

$[asy] pair A=(0,0); pair B=(0,2); pair C=(1,1); draw(A--B--C); draw(A--C--B); [/asy]$

$\text{Homework: calculate the area}$

ANSWER: LOOK ONLY WHEN THE PROBLEM IS COMPLETED >>> THIS IS ONLY TO CHECK YOUR ANSWER

$\text{This is an isosceles right triangle. It has side length }\sqrt{2}$

$\text{It has an height of } \frac{\sqrt{2}}{\sqrt{2}} = 1 \text{ and a base of } \sqrt{2} \cdot \sqrt{2} = 2$

$\text{The area is } 1 \cdot 2 \cdot \frac{1}{2} = \boxed{1}$ +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

Pythagorean

$[asy] pair A=(0,0); pair B=(0,3); pair C=(4,0); draw(A--B--C--A,black+linewidth(1)); pair D=(-3,0); pair E=(-3,3); draw(A--B--E--D--A,red+linewidth(1)); pair F=(0,-4); pair G=(4,-4); draw(A--C--G--F--A,orange+linewidth(1)); pair H=(7,4); pair I=(3,7); draw(B--C--H--I--B,yellow+linewidth(1)); draw(A--B--C--A,black+linewidth(1)); [/asy]$ $\text{The pythagorean theorem: red plus orange equals yellow}$

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

Isosceles Trapezoid

$[asy] pair A=(0,0); pair B=(10,0); pair C=(6,8); pair D=(2,8); pair E=(30/7,40/7); draw(A--B--C--D--A,black+linewidth(5)); fill(A--E--D--cycle,red); fill(B--E--C--cycle,yellow); dot(A); dot(B); dot(C); dot(D); dot(E); label(A,'A',SW); label(B,'B',SE); label(C,'C',NE); label(D,'D',NW); label(E,'E',S); draw(A--E--C,blue+linewidth(2)); draw(B--E--D,blue+linewidth(2)); [/asy]$ $\text{Given that } \overline{AB} \parallel \overline{DC} \text{ , prove that } A_\triangle AED = A_\triangle BEC .$

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

Hexagon (In AMC 2023 10A #24)

$[asy] unitsize(5cm); draw(scale(3)*polygon(6)); filldraw(shift(dir(0)*2+dir(120)*3/7)*polygon(6), lightgray); filldraw(shift(dir(60)*2+dir(180)*3/7)*polygon(6), lightgray); filldraw(shift(dir(120)*2+dir(240)*3/7)*polygon(6), lightgray); filldraw(shift(dir(180)*2+dir(300)*3/7)*polygon(6), lightgray); filldraw(shift(dir(240)*2+dir(0)*3/7)*polygon(6), lightgray); filldraw(shift(dir(300)*2+dir(60)*3/7)*polygon(6), yellow); pair A = (0,0) + 3 * dir(300); pair B = A + 3/7 *dir(60); pair C = B + 1 * dir(180); pair D = C + 3/7 * dir(240); pair E = C + 4/7 * dir(120); pair F = E + 3/7 * dir(240); pair G = F + 4/7 * dir(240); pen p = red+linewidth(6); draw(A--B--C--D--cycle,p); draw(C--E--F--D--cycle,p); draw(F--G--D--cycle,p); label("1",(B + C)/2,dir(90)); label("1",(A + D)/2,dir(270)); label("3/7",(A + B)/2,dir(330)); label("3/7",(C + D)/2,dir(330)); label("4/7",(C+E)/2,dir(30)); label("3/7",(E+F)/2,dir(150)); label("4/7",(F+D)/2,dir(30)); label("4/7",(F+G)/2,dir(150)); label("4/7",(G+D)/2,dir(270)); [/asy]$

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

Tiny 2D experiments

Grid with numbers

$[asy] unitsize(36); pair A1 = (0,0); pair A2 = (0,5); pair A3 = (5,5); pair A4 = (5,0); fill(A1--A2--A3--A4--cycle,gray(0.61437908496)); draw(A1--A4--A4+dir(90)--A1+dir(90)--A1+dir(90)*2--A4+dir(90)*2--A3-dir(90)*2--A2-dir(90)*2--A2-dir(90)--A3-dir(90)--A3--A2,linewidth(3)); draw(A1--A2--A2+dir(0)--A1+dir(0)--A1+dir(0)*2--A2+dir(0)*2--A3-dir(0)*2--A4-dir(0)*2--A4-dir(0)--A3-dir(0)--A3--A4,linewidth(3)); pair B1 = (0,1/2); pair B2 = (0,3/2); pair B3 = (0,5/2); pair B4 = (0,7/2); pair B5 = (0,9/2); pair C1 = (1/2,0); pair C2 = (3/2,0); pair C3 = (5/2,0); pair C4 = (7/2,0); pair C5 = (9/2,0); label("0",B1+C1); label("0",B2+C1); label("0",B3+C1); label("0",B4+C1); label("0",B5+C1); label("0",B1+C2); label("0",B2+C2); label("0",B3+C2); label("1",B4+C2); label("1",B5+C2); label("0",B1+C3); label("0",B2+C3); label("2",B3+C3); label("1",B4+C3); label("0",B5+C3); label("0",B1+C4); label("5",B2+C4); label("3",B3+C4); label("0",B4+C4); label("0",B5+C4); label("7",B1+C5); label("1",B2+C5); label("0",B3+C5); label("2",B4+C5); label("0",B5+C5); [/asy]$

Tiny 3D experiments

Cube

$[asy] unitsize(60); pair A = (0,0); pair B = A + dir(0); pair C = A + dir(60); pair D = A + dir(120); pair E = A + dir(180); pair F = A + dir(240); pair G = A + dir(300); filldraw(A--B--C--D--cycle,orange); filldraw(A--D--E--F--cycle,blue); filldraw(A--F--G--B--cycle,gray); [/asy]$

Smaller cube

$[asy] unitsize(40); pair A = (0,0); pair B = A + dir(0); pair C = A + dir(60); pair D = A + dir(120); pair E = A + dir(180); pair F = A + dir(240); pair G = A + dir(300); filldraw(A--B--C--D--cycle,orange); filldraw(A--D--E--F--cycle,blue); filldraw(A--F--G--B--cycle,gray); [/asy]$

Cube with a cube hole in it

$[asy] unitsize(60); pair A = (0,0); pair B = A + dir(0); pair b = A + 2/3*dir(0); pair C = A + dir(60); pair c = A + 2/3*dir(60); pair D = A + dir(120); pair d = A + 2/3*dir(120); pair E = A + dir(180); pair e = A + 2/3*dir(180); pair F = A + dir(240); pair f = A + 2/3*dir(240); pair G = A + dir(300); pair g = A + 2/3*dir(300); filldraw(A--e--f--g--cycle,orange); filldraw(A--g--b--c--cycle,blue); filldraw(A--c--d--e--cycle,gray); filldraw(b--c--d--D--C--B--cycle,orange); filldraw(d--e--f--F--E--D--cycle,blue); filldraw(f--g--b--B--G--F--cycle,gray); [/asy]$

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

Rotating cube

Perspective

$[asy] import three; unitsize(3); currentprojection=perspective(2,0,1/2); draw((0,0,0)--(1,0,0)--(1,1,0)--(0,1,0)--cycle,red); draw((0,0,0)--(0,0,1),red); draw((0,1,0)--(0,1,1),red); draw((1,1,0)--(1,1,1),red); draw((1,0,0)--(1,0,1),red); draw((0,0,1)--(1,0,1)--(1,1,1)--(0,1,1)--cycle,red); [/asy]$

$[asy] import three; unitsize(3); currentprojection=perspective(2*Cos(15),2*Sin(15),1/2); draw((0,0,0)--(1,0,0)--(1,1,0)--(0,1,0)--cycle,red); draw((0,0,0)--(0,0,1),red); draw((0,1,0)--(0,1,1),red); draw((1,1,0)--(1,1,1),red); draw((1,0,0)--(1,0,1),red); draw((0,0,1)--(1,0,1)--(1,1,1)--(0,1,1)--cycle,red); [/asy]$

$[asy] import three; unitsize(3); currentprojection=perspective(2*Cos(30),2*Sin(30),1/2); draw((0,0,0)--(1,0,0)--(1,1,0)--(0,1,0)--cycle,red); draw((0,0,0)--(0,0,1),red); draw((0,1,0)--(0,1,1),red); draw((1,1,0)--(1,1,1),red); draw((1,0,0)--(1,0,1),red); draw((0,0,1)--(1,0,1)--(1,1,1)--(0,1,1)--cycle,red); [/asy]$

$[asy] import three; unitsize(3); currentprojection=perspective(2*Cos(45),2*Sin(45),1/2); draw((0,0,0)--(1,0,0)--(1,1,0)--(0,1,0)--cycle,red); draw((0,0,0)--(0,0,1),red); draw((0,1,0)--(0,1,1),red); draw((1,1,0)--(1,1,1),red); draw((1,0,0)--(1,0,1),red); draw((0,0,1)--(1,0,1)--(1,1,1)--(0,1,1)--cycle,red); [/asy]$

$[asy] import three; unitsize(3); currentprojection=perspective(2*Cos(60),2*Sin(60),1/2); draw((0,0,0)--(1,0,0)--(1,1,0)--(0,1,0)--cycle,red); draw((0,0,0)--(0,0,1),red); draw((0,1,0)--(0,1,1),red); draw((1,1,0)--(1,1,1),red); draw((1,0,0)--(1,0,1),red); draw((0,0,1)--(1,0,1)--(1,1,1)--(0,1,1)--cycle,red); [/asy]$

$[asy] import three; unitsize(3); currentprojection=perspective(2*Cos(75),2*Sin(75),1/2); draw((0,0,0)--(1,0,0)--(1,1,0)--(0,1,0)--cycle,red); draw((0,0,0)--(0,0,1),red); draw((0,1,0)--(0,1,1),red); draw((1,1,0)--(1,1,1),red); draw((1,0,0)--(1,0,1),red); draw((0,0,1)--(1,0,1)--(1,1,1)--(0,1,1)--cycle,red); [/asy]$

Orthographic

$[asy] import three; unitsize(3); currentprojection=orthographic(2,0,1/2); draw((0,0,0)--(1,0,0)--(1,1,0)--(0,1,0)--cycle,red); draw((0,0,0)--(0,0,1),red); draw((0,1,0)--(0,1,1),red); draw((1,1,0)--(1,1,1),red); draw((1,0,0)--(1,0,1),red); draw((0,0,1)--(1,0,1)--(1,1,1)--(0,1,1)--cycle,red); [/asy]$

$[asy] import three; unitsize(3); currentprojection=orthographic(2*Cos(15),2*Sin(15),1/2); draw((0,0,0)--(1,0,0)--(1,1,0)--(0,1,0)--cycle,red); draw((0,0,0)--(0,0,1),red); draw((0,1,0)--(0,1,1),red); draw((1,1,0)--(1,1,1),red); draw((1,0,0)--(1,0,1),red); draw((0,0,1)--(1,0,1)--(1,1,1)--(0,1,1)--cycle,red); [/asy]$

$[asy] import three; unitsize(3); currentprojection=orthographic(2*Cos(30),2*Sin(30),1/2); draw((0,0,0)--(1,0,0)--(1,1,0)--(0,1,0)--cycle,red); draw((0,0,0)--(0,0,1),red); draw((0,1,0)--(0,1,1),red); draw((1,1,0)--(1,1,1),red); draw((1,0,0)--(1,0,1),red); draw((0,0,1)--(1,0,1)--(1,1,1)--(0,1,1)--cycle,red); [/asy]$

$[asy] import three; unitsize(3); currentprojection=orthographic(2*Cos(45),2*Sin(45),1/2); draw((0,0,0)--(1,0,0)--(1,1,0)--(0,1,0)--cycle,red); draw((0,0,0)--(0,0,1),red); draw((0,1,0)--(0,1,1),red); draw((1,1,0)--(1,1,1),red); draw((1,0,0)--(1,0,1),red); draw((0,0,1)--(1,0,1)--(1,1,1)--(0,1,1)--cycle,red); [/asy]$

$[asy] import three; unitsize(3); currentprojection=orthographic(2*Cos(60),2*Sin(60),1/2); draw((0,0,0)--(1,0,0)--(1,1,0)--(0,1,0)--cycle,red); draw((0,0,0)--(0,0,1),red); draw((0,1,0)--(0,1,1),red); draw((1,1,0)--(1,1,1),red); draw((1,0,0)--(1,0,1),red); draw((0,0,1)--(1,0,1)--(1,1,1)--(0,1,1)--cycle,red); [/asy]$

$[asy] import three; unitsize(3); currentprojection=orthographic(2*Cos(75),2*Sin(75),1/2); draw((0,0,0)--(1,0,0)--(1,1,0)--(0,1,0)--cycle,red); draw((0,0,0)--(0,0,1),red); draw((0,1,0)--(0,1,1),red); draw((1,1,0)--(1,1,1),red); draw((1,0,0)--(1,0,1),red); draw((0,0,1)--(1,0,1)--(1,1,1)--(0,1,1)--cycle,red); [/asy]$

Circle diagrams

A collection of stuff having to do with circles

Intersection points

$[asy] unitsize(9); pair A1 = (0,0); pair A2 = A1 + dir(0)*30; pair A3 = A2 + dir(0)*30; pair B1 = (9,0); pair B2 = B1 + dir(0)*30; pair B3 = B2 + dir(0)*30; pair G1 = (9/2,-15/2); pair G2 = G1 + dir(0)*30; pair G3 = G2 + dir(0)*30; pair H1 = A1 + dir(90)*5; pair H2 = A2 + dir(90)*5; pair H3 = A3 + dir(90)*5; pair I1 = B1 + dir(90)*3; pair I2 = B2 + dir(90)*4; pair I3 = B3 + dir(90)*5; pair J1 = A1 + dir(120)*5; pair J2 = A2 + dir(120)*5; pair J3 = A3 + dir(120)*5; pair K1 = B1 + dir(60)*3; pair K2 = B2 + dir(60)*4; pair K3 = B3 + dir(60)*5; pair N1 = G1 + dir(270)*3; pair N2 = G2 + dir(270)*3; pair N3 = G3 + dir(270)*3; path E1 = circle(A1,5); path E2 = circle(A2,5); path E3 = circle(A3,5); path F1 = circle(B1,3); path F2 = circle(B2,4); path F3 = circle(B3,5); path L1 = A1--J1; path L2 = A2--J2; path L3 = A3--J3; path M1 = B1--K1; path M2 = B2--K2; path M3 = B3--K3; path O1 = A1--B1; path O2 = A2--B2; path O3 = A3--B3; draw(E1,red); draw(E2,red); draw(E3,red); draw(F1,green); draw(F2,green); draw(F3,green); draw(L1,red); draw(L2,red); draw(L3,red); draw(M1,green); draw(M2,green); draw(M3,green); draw(O1,blue); draw(O2,blue); draw(O3,blue); dot(A1); dot(A2); dot(A3); dot(B1); dot(B2); dot(B3); label("$O_{A1}$",A1,dir(240),red); label("$O_{A2}$",A2,dir(240),red); label("$O_{A3}$",A3,dir(240),red); label("$O_{B1}$",B1,dir(300),green); label("$O_{B2}$",B2,dir(300),green); label("$O_{B3}$",B3,dir(300),green); label("No points",G1); label("One point",G2); label("Two points",G3); label("$r_{A1}+r_{B1}<d_1$",N1); label("$r_{A2}+r_{B2}=d_2$",N2); label("$r_{A3}+r_{B3}>d_3$",N3); label("$r_{A1}$",L1,dir(210),red); label("$r_{A2}$",L2,dir(210),red); label("$r_{A3}$",L3,dir(210),red); label("$r_{B1}$",M1,dir(330),green); label("$r_{B2}$",M2,dir(330),green); label("$r_{B3}$",M3,dir(330),green); label("$d_1$",O1,dir(240),blue); label("$d_2$",O2,dir(240),blue); label("$d_3$",O3,dir(240),blue); pair [] D1 = intersectionpoints(E2,F2); pair [] D2 = intersectionpoints(E3,F3); pair C1 = D1[0]; pair C2 = D2[0]; pair C3 = D2[1]; dot(C1,blue); dot(C2,blue); dot(C3,blue); label("$I_2$",C1,dir(315),blue); label("$I_{3a}$",C2,dir(0),blue); label("$I_{3b}$",C3,dir(0),blue); [/asy]$

Created files

(Using desmos)

Colored dodecagon

Original file:

File

Pentagon 2023 12B Q25 dissmo

Original size:

Page

Toolbox

Search

User:Afly

Contents

Afly's notes

About Afly

Some of my asymptote creations

Triangle

Pythagorean

Isosceles Trapezoid

Hexagon (In AMC 2023 10A #24)

Tiny 2D experiments

Grid with numbers

Tiny 3D experiments

Cube

Smaller cube

Cube with a cube hole in it

Rotating cube

Perspective

Orthographic

Circle diagrams

Intersection points

Created files

Colored dodecagon

Pentagon 2023 12B Q25 dissmo

Yrock's Notes

ConfidentFlamingo's notes

Other Users' notes