Solving Olympiad Level Geometry Problems with Complex Numbers

We thank Patrick Bauermann and Karl Fegert for their valuable feedback and for their permission to use the official logo of the Bundeswettbewerb Mathematik.

0:00 Introduction
1:32 Problem
2:20 Basics
5:42 Complex Plane
7:24 Addition and Subtraction
8:30 Scaling
9:30 Lines
11:10 Intersection of Lines
12:15 Absolute Value and Argument
13:10 Multiplication and Division
14:30 Rotation
15:43 Without Loss of Generality
19:33 Solution - Part 1
24:32 Solution - Part 2
28:06 Further Reading

Corrections:
9:20 the second -lambda should be replaced with +lambda
13:09 we meant to say: "The astute scholars amongst you may have noticed that we just looked at Cartesian geometry up to this point."
24:53 we meant to say: "We just know that D is somewhere on the line segment AC."

Resources from this video (including the proof): https://github.com/mathelehrer/geomet...
Resources from the Swiss Math Olympiad: https://mathematical.olympiad.ch/de/s...
Resources from the Bundeswettbewerb Mathematik: https://www.mathe-wettbewerbe.de/aufg...

This video is a submission for the Summer of Math Exposition 2:    • Summer of Math Exposition 2 Invitation  
It is a collaboration between Alexandru Duca (the domain expert) and me (the content producer).
You can find our original pitches on GitHub:
Alexandru Duca: https://github.com/leios/SoME_Topics/...
Number Cruncher: https://github.com/leios/SoME_Topics/...

While writing the script, Alexandru Duca drew inspiration from the following videos:
"The hardest problem on the hardest test" by 3Blue1Brown:    • The hardest problem on the hardest test  
"Vectors | Chapter 1, Essence of linear algebra" by 3Blue1Brown:    • Vectors | Chapter 1, Essence of linear alg...   ‪@3blue1brown‬
"String Theory Explained – What is The True Nature of Reality?" by Kurzgesagt:    • String Theory Explained – What is The True...   ‪@kurzgesagt‬

I learned to make some of the animations from the following videos:
Curtain:    • How to make REALISTIC CURTAIN in 3 ways - ...   ‪@QuyTran37‬
Sand Texture:    • Easy Procedural Sand Material -  Blender 2...   ‪@TR_MediaOfficial‬
Engraved Text:    • Engrave & Emboss Text Easily In Blender | ...   ‪@5MinutesBlender‬
Rope:    • Blender 3.1  Rigid Body And Cloth Simulati...   ‪@mk3dgraphics‬
Marble Texture:    • Procedural Marble Material (Blender Tutorial)   ‪@RyanKingArt‬
Gold Texture:    • Procedural Gold Material (Blender Tutorial)  
Explosions:    • VFX - Avengers Infinity War Dust Effect in...   ‪@3d-illusions‬
Flag:    • FLAGS (with blender)   ‪@CGMatter‬
Cell Fracture Explosion:    • Blender 2.8 Tutorial - Cell Fracture Explo...   ‪@blendermania3d‬

Music by Chris Zabriskie: http://chriszabriskie.com/
"Candlepower": http://chriszabriskie.com/divider/
"CGI Snake": http://chriszabriskie.com/divider/
"Divider": http://chriszabriskie.com/divider/
"I Am Running Down the Long Hallway of Viewmont Elementary": http://chriszabriskie.com/honor/
"We Always Thought the Future Would Be Kind of Fun": http://chriszabriskie.com/darkglow/
"What True Self, Feels Bogus, Let's Watch Jason X": http://chriszabriskie.com/reappear/
Licensed under Creative Commons Attribution 4.0: https://creativecommons.org/licenses/...