Ceva’s Theorem Is More Than a Formula for Concurrency

Submitted by Kevin rain on

In triangle geometry, one of the most efficient ways to test whether three cevians are concurrent is Ceva’s Theorem.

Let D, E, F lie on sides BC, CA, AB respectively of triangle ABC. Then lines AD, BE, and CF are concurrent if and only if

(BD/DC) * (CE/EA) * (AF/FB) = 1.

What I find interesting is that many students learn this as a “contest trick,” but it is actually a very natural statement. The theorem says that concurrency is encoded by a balance condition on the three side partitions.

Cycloidal Curves: Theory, Parametric Equations, and Animation

Jhun Vert
Cycloid

The purpose of this post is to systematically derive the corresponding parametric equations and implement them in Python for visualization using Manim. The goal is not only to understand these curves analytically but also to see them come alive through animation.

cardioid_fixed_visible_800px.gif
Cardioid ($r = R$), a special case of the epicycloid.

 

The Intuition Behind Integration by Parts (Proof & Example)

Submitted by Kevin rain on

Many students simply memorize the Integration by Parts formula: ∫ u dv = uv - ∫ v du, but understanding where it comes from makes it much easier to remember and apply during exams.

1. The Intuitive Proof (Reverse Product Rule)

The entire concept is actually just the Product Rule for differentiation in reverse.
From the product rule for two differentiable functions, u and v: (uv)' = u'v + uv'