重心座標簡介
這份筆記介紹了一個簡單但功能強大的數學工具，常用於數學奧林匹亞競賽中 —— 重心座標(barycentric coordinates)。重心座標是一種齊次座標，也就是說，$[a : b : c]$ 和 $[ka : kb : kc]$（只要 $k \neq 0$）表示的是相同的點。
重心座標是以一個參考三角形為基礎定義的。假設 $\triangle ABC$ 是一個固定的三角形，則對於任意一點 $P$，其重心座標定義為 \[ P = \left[\frac{[\triangle PBC]}{[\triangle ABC]}, \frac{[\triangle PCA]}{[\triangle BCA]}, \frac{[\triangle PAB]}{[\triangle CAB]}\right], \] 其中，$[\triangle XYZ]$ 表示 $\triangle XYZ$ 的有向面積。
利用重心座標，可以推導出許多幾何性質簡潔的方程式，例如直線的方程式、垂直條件的判斷、圓的方程式等。完整的內容詳見附上的 PDF 文件，但此份筆記以中文撰寫。
Introduction to Barycentric Coordinate
This note introduces a simple yet powerful tool frequently used in mathematical olympiads —— barycentric coordinates. Barycentric coordinates are a type of homogeneous coordinate, meaning that the points represented by $[a : b : c]$ and $[ka : kb : kc]$ are equivalent as long as $k \neq 0$.
Barycentric coordinates are defined relative to a reference triangle. Suppose $\triangle ABC$ is a fixed triangle. For any point $P$, its barycentric coordinates are given by \[ \left[\frac{[\triangle PBC]}{[\triangle ABC]},\frac{[\triangle PCA]}{[\triangle BCA]},\frac{[\triangle PAB]}{[\triangle CAB]} \right], \] where $[\triangle XYZ]$ represents the signed area of $\triangle XYZ$.
Using barycentric coordinates, we can derive elegant expressions for various geometric properties and conditions, such as the equations of lines, the criteria for perpendicularity, and the equations of circles, among others.
For a detailed explanation, please refer to the full note in the provided PDF. Note that the document is written in Mandarin Chinese.