三角形

考什么 · What it tests

需要先会 · Prerequisites

• 内角和 $180°$ / The $180°$ angle sum
• 面积 $=\frac{1}{2}\times\text{底}\times\text{高}$ / Area $=\frac{1}{2}\times\text{base}\times\text{height}$
• 平方与开方（勾股要用） / Squaring and square roots (for the Pythagorean theorem)

常见套路 · Common moves

1. 相似定比例，比例定长度：两三角形相似时对应边成比例，一个比例式就能求出未知边——高星几何最常用的招。

   Similarity gives ratios, ratios give lengths: in similar triangles corresponding sides are proportional, so one proportion recovers an unknown side — the go-to move on harder geometry.
2. 勾股凑特殊数：见直角先试常见勾股数 3-4-5、5-12-13、8-15-17，往往不用真开方。

   Look for Pythagorean triples: at a right angle, try the familiar 3-4-5, 5-12-13, 8-15-17 first — often you skip taking a root at all.
3. 等面积换底高：同一个三角形面积不变，换一条底边就能反求对应的高，常用来求点到线的距离。

   Swap base and height at fixed area: a triangle's area is the same whichever side is the base, so switching bases recovers a height — handy for a point-to-line distance.

易错点 · Common pitfalls

• 相似比直接当面积比：面积比是相似比的平方，长度比 $2:3$ 对应面积比 $4:9$。

  Using the similarity ratio as the area ratio: the area ratio is its square, so a length ratio $2:3$ gives an area ratio $4:9$.
• 勾股用错边：斜边（最长边、对着直角）必须单独在等式一边，$a^2+b^2=c^2$ 里 $c$ 是斜边。

  Misplacing the sides in the Pythagorean theorem: the hypotenuse (longest side, opposite the right angle) stands alone, so $c$ in $a^2+b^2=c^2$ is the hypotenuse.