Mastering Complex Numbers - Proving Ptolemy's Inequality with Complex Numbers and Algebra

In this episode, we unveil the beautiful interplay between complex numbers, algebra, and geometry.

Our journey will revolve around proving a classical geometric inequality for a quadrilateral: AB x CD + AD x BC ≥ AC x BD. This inequality is often tackled using traditional geometric approaches, but we'll bring a fresh perspective by using complex numbers and algebra.

We'll demonstrate how to interpret the difference of complex numbers geometrically, and apply the triangle inequality in the complex plane, coupled with some clever algebraic manipulations. This approach provides a novel and powerful method for handling geometric inequalities.

0:00 Problem Introduction
1:03 Complex Numbers: Geometric Interpretation of Difference
1:43 Formulate the Geometric Inequality as A Complex Number Inequality
4:04 Triangle Inequality for Complex Numbers
4:54 Algebra Tricks with z + w
5:50 The Proof
6:30 When Does the Equality Hold? Concyclic

Whether you're a student, a teacher, or simply someone who loves math, this video offers a fascinating perspective, shedding new light on the profound connections within mathematics.

Join us in this exciting exploration, and deepen your understanding and mastery of complex numbers.

[Alex-xumo]

You said at the end that they have to be concycling that is be on the circle, but what about angle D which is not on the circle?
I suppose inequalitybstill holds but last point you made about them needing to be on a circle I am not able to follow

[MochiaoChen]

emmmm I cannot find your email...so i write here
Try this one, the common believed hardest gaokao math problem
for xyz=8, x, y, z>0, prove