Key Theorems in Plane Geometry and Their Applications | 平面几何常用定理及其应用

📚 Key Theorems in Plane Geometry and Their Applications | 平面几何常用定理及其应用

Plane geometry is built upon a small set of fundamental theorems. Mastering these theorems and their logical connections is essential for solving problems efficiently, whether in exams or real-world design. This article reviews the most frequently used theorems, explains their proofs briefly, and demonstrates how to apply them in typical exercises.

平面几何建立在一小撮基本定理之上。掌握这些定理及其逻辑联系，是高效解题的关键，无论面对考试还是实际设计。本文将梳理最常用的定理，简要说明其证明，并通过典型例题演示它们的应用。

1. Triangle Angle Sum and Pythagoras’ Theorem | 三角形内角和与勾股定理

The sum of the interior angles of a triangle is always 180°. This is the most basic property in plane geometry. If two angles of a triangle are known, the third can be found immediately. For a right triangle, Pythagoras’ Theorem states that the square of the hypotenuse equals the sum of the squares of the other two sides:

三角形内角和恒为180°，这是平面几何中最基本的性质。若已知三角形中两个角，第三个角立即可求。对于直角三角形，勾股定理指出斜边的平方等于两直角边的平方和：

a² + b² = c²

The converse of Pythagoras’ Theorem is equally useful: if the three sides of a triangle satisfy a² + b² = c², then the angle opposite the side c is a right angle.

勾股定理的逆定理同样重要：若三角形三边满足 a² + b² = c²，则 c 边所对的角是直角。

Example: In triangle ABC, AB = 3, BC = 4, AC = 5. Since 3² + 4² = 9 + 16 = 25 = 5², angle B is 90°.

例：在三角形 ABC 中，AB = 3，BC = 4，AC = 5。因为 3² + 4² = 9 + 16 = 25 = 5²，所以角 B 为90°。

2. Congruent Triangles | 全等三角形的判定

Two triangles are congruent when their corresponding sides and angles are equal. The common congruence criteria are: SSS (three sides equal), SAS (two sides and the included angle equal), ASA (two angles and the included side equal), and AAS (two angles and a non-included side equal). For right triangles, HL (hypotenuse and one leg equal) is also a valid criterion.

当两个三角形的对应边和对应角都相等时，它们全等。常用的全等判定有：SSS（三边相等）、SAS（两边及其夹角相等）、ASA（两角及其夹边相等）、AAS（两角及其中一角的对边相等）。对于直角三角形，HL（斜边和一条直角边相等）也是有效判定。

Congruent triangles are used to prove equal lengths or equal angles by showing that two triangles are congruent. This is often achieved by adding auxiliary lines such as midpoints or perpendiculars.

全等三角形常用于证明线段相等或角相等，通过证明两个三角形全等来实现。通常需要添加辅助线，如取中点或作垂线。

Example: In triangle ABC, AB = AC, and AD is the median from A to BC. Then triangles ABD and ACD are congruent by SSS, so AD is also the altitude and angle bisector.

例：在三角形 ABC 中，AB = AC，AD 是 BC 边上的中线。则三角形 ABD 与 ACD 满足 SSS 全等，因此 AD 同时也是高和角平分线。

3. Similar Triangles | 相似三角形的判定

Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional. The similarity criteria are AA (two angles equal), SAS (two sides proportional and the included angle equal), and SSS (three sides proportional). Once similarity is established, ratios of corresponding sides yield unknown lengths.

若两个三角形的对应角相等、对应边成比例，则它们相似。相似判定有：AA（两角相等）、SAS（两边成比例且夹角相等）、SSS（三边成比例）。一旦确定相似，由对应边的比例即可求未知长度。

A classic application is the basic proportionality theorem (Thales’ theorem): if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.

经典应用是基本比例定理（泰勒斯定理）：平行于三角形一边的直线截另外两边，所得的对应线段成比例。

If DE ∥ BC, then AD/DB = AE/EC

若 DE ∥ BC，则 AD/DB = AE/EC

Example: In triangle ABC, D is on AB with AD:DB = 2:3, and DE ∥ BC meeting AC at E. Then AE:EC = 2:3.

例：在三角形 ABC 中，D 在 AB 上，AD:DB = 2:3，DE ∥ BC 交 AC 于 E。则 AE:EC = 2:3。

4. Angle Bisector Theorem and Midsegment Theorem | 角平分线定理与中位线定理

The angle bisector theorem states that the angle bisector of an angle in a triangle divides the opposite side into segments proportional to the adjacent sides:

角平分线定理：三角形一个角的平分线将对边分成两条线段，这两条线段与该角的两邻边成比例：

BD/DC = AB/AC

Here AD is the angle bisector of ∠A, meeting BC at D. This theorem is extremely useful for finding side lengths when the angle bisector is present.

其中 AD 是∠A 的平分线，交 BC 于 D。这个定理在已知角平分线时求边长非常有用。

The midsegment theorem: the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length.

中位线定理：三角形两边中点的连线平行于第三边，且长度等于第三边的一半。

DE = ½BC and DE ∥ BC

DE = ½BC 且 DE ∥ BC

These two theorems often work together in problems involving quadrilaterals, especially when midpoints are given.

这两个定理在涉及四边形的题目中常联合使用，尤其当给出中点条件时。

5. Perpendicular Diameter Theorem (Chord Properties) | 垂径定理与弦的性质

In a circle, if a diameter is perpendicular to a chord, then it bisects the chord and the arcs subtended by the chord. Conversely, the perpendicular bisector of any chord passes through the centre of the circle.

在圆中，若一条直径垂直于弦，则它平分这条弦及其所对的两条弧。反过来，任意弦的垂直平分线必过圆心。

This theorem enables calculations of chord length, distance from the centre to a chord, and radius. Given radius r, distance d from centre to chord, and chord length l, they satisfy:

该定理可用于计算弦长、弦心距和半径。设半径为 r，弦心距为 d，弦长为 l，三者满足：

(l/2)² + d² = r²

Example: A chord of length 8 is at distance 3 from the centre. Then r = √(4² + 3²) = 5.

例：一弦长8，弦心距为3。则半径 r = √(4² + 3²) = 5。

The equal-chord theorem also states: equal chords subtend equal arcs and equal central angles; chords with equal distance from the centre are equal.

等弦定理还指出：等弦所对的弧相等、所对的圆心角相等；弦心距相等的弦长度相等。

6. Inscribed Angle Theorem | 圆周角定理

The measure of an inscribed angle is half the measure of its intercepted central angle. Thus, in a circle, angles subtended by the same arc are equal. This is one of the most powerful tools in circle geometry.

圆周角定理：圆周角的度数等于它所对弧上的圆心角度数的一半。因此，同弧所对的圆周角相等。这是圆几何中最有力的工具之一。

θ = ½ × central angle

θ = ½ × 圆心角

A special result: the angle subtended by a diameter is a right angle (Thales’ theorem). If triangle ABC is inscribed in a circle and AC is a diameter, then ∠ABC = 90°.

特别地：直径所对的圆周角是直角（泰勒斯定理）。若三角形 ABC 内接于圆，且 AC 为直径，则∠ABC = 90°。

Also, opposite angles of a cyclic quadrilateral are supplementary:

此外，圆内接四边形的对角互补：

∠A + ∠C = 180°, ∠B + ∠D = 180°

∠A + ∠C = 180°，∠B + ∠D = 180°

This property often converts an angle problem into an algebra equation.

这个性质常将角度问题转化为代数方程。

7. Tangent Theorems | 切线的性质与判定

A tangent to a circle is perpendicular to the radius at the point of tangency. Conversely, if a line through a point on the circle is perpendicular to the radius at that point, then it is a tangent.

圆的切线垂直于过切点的半径；反之，过圆上一点且垂直于该点半径的直线是切线。

The tangent-secant theorem (power of a point) is also fundamental: from an external point P, the tangent length PT satisfies PT² = PA × PB, where PAB is a secant intersecting the circle at A and B.

切割线定理（圆幂定理）同样基础：从圆外一点 P 引切线 PT，则 PT² = PA × PB，其中 PAB 为割线，交圆于 A、B 两点。

PT² = PA · PB

The tangent length theorem states that tangent segments drawn from the same external point are equal. This is useful in solving problems with two tangents from one point.

切线长定理：从圆外一点引圆的两条切线，它们的长度相等。这在处理过同一点的两条切线问题时非常有用。

Example: From point P outside a circle, tangent PA = 6, secant through P intersects the circle at B and C with PB = 4. Then PC = 6² / 4 = 9.

例：圆外一点 P，切线 PA = 6，过 P 的割线交圆于 B、C，且 PB = 4。则 PC = 6² / 4 = 9。

8. Cyclic Quadrilateral and Ptolemy’s Theorem | 圆内接四边形与托勒密定理

A quadrilateral is cyclic if all four vertices lie on a circle. A common test for concyclicity: if two opposite angles sum to 180°, the quadrilateral is cyclic. Also, an exterior angle of a cyclic quadrilateral equals its opposite interior angle.

若四边形的四个顶点都在同一个圆上，则它是圆内接四边形。判断四点共圆的一个常用条件：一对对角之和为180°。此外，圆内接四边形的外角等于它的内对角。

Ptolemy’s Theorem relates the sides and diagonals of a cyclic quadrilateral:

托勒密定理给出了圆内接四边形的边与对角线之间的关系：

AC × BD = AB × CD + BC × AD

This theorem is particularly elegant for finding the length of a diagonal when the four sides are known.

当已知四边长度时，这个定理可优雅地求对角线的长度。

In an equilateral triangle inscribed in a circle, Ptolemy’s theorem can prove the simple relation between a point on the arc and the vertices.

在圆内接等边三角形的背景下，托勒密定理还可用来证明弧上一点到三顶点距离之间的简单关系。

9. Ceva’s Theorem and Menelaus’ Theorem | 塞瓦定理与梅涅劳斯定理

Ceva’s theorem gives a criterion for three cevians (lines from vertices to opposite sides) to be concurrent. In triangle ABC, points D on BC, E on CA, F on AB satisfy:

塞瓦定理给出了三角形中三条塞瓦线（从顶点到对边的直线）共点的条件。在三角形 ABC 中，D 在 BC 上，E 在 CA 上，F 在 AB 上，它们满足：

(BD/DC) × (CE/EA) × (AF/FB) = 1

This holds if and only if AD, BE, CF are concurrent. It is a powerful tool for proving collinearity and concurrency of lines.

这个等式成立当且仅当 AD、BE、CF 三线共点。它是证明三线共点或点在线上问题的强大工具。

Menelaus’ theorem describes collinearity of three points on the sides of a triangle (extended as needed). For points D on BC, E on CA, F on AB, they are collinear if:

梅涅劳斯定理则描述三角形三边（或延长线）上三个点共线的条件。对于 D 在 BC 上，E 在 CA 上，F 在 AB 上，它们共线当且仅当：

(BD/DC) × (CE/EA) × (AF/FB) = -1

Here directed segments are used, so the product is -1. In many olympiad problems, Menelaus’ theorem quickly proves collinearity that would otherwise be difficult to show.

其中采用有向线段，因此乘积为 -1。在竞赛题中，梅涅劳斯定理能快速证明共线，而其他方法可能相当困难。

10. Comprehensive Application and Problem-Solving Strategy | 综合应用与解题策略

When tackling a plane geometry problem, first identify which theorems are likely involved: if midpoints appear, consider the midsegment theorem; if perpendicular bisectors appear, recall chord properties; if angle bisectors appear, apply the angle bisector theorem. Drawing auxiliary lines is a critical skill.

解决平面几何问题时，首先判断可能涉及哪些定理：出现中点考虑中位线定理；出现垂直平分线联想弦的性质；出现角平分线则应用角平分线定理。添加辅助线是关键技能。

Worked example: In circle O, diameter AB is 10, chord CD is 8, and CD is parallel to AB. Find the distance between AB and CD.

例题：在圆 O 中，直径 AB 为10，弦 CD 为8，且 CD ∥ AB。求 AB 与 CD 之间的距离。

Since AB is a diameter, the radius is 5. By the perpendicular diameter theorem, the distance from the centre to CD is √(5² – 4²) = 3. Because CD is parallel to AB, the distance between them is exactly 3.

因为 AB 是直径，半径为5。由垂径定理，圆心到 CD 的距离为 √(5² – 4²) = 3。由于 CD ∥ AB，因此两弦之间的距离就是3。

Strategy table:

策略对照表：

Condition / 条件	Likely Theorem / 常用定理
Midpoint of two sides / 两边中点	Midsegment theorem / 中位线定理
Angle bisector / 角平分线	Angle bisector theorem / 角平分线定理
Diameter in a circle / 圆中直径	Inscribed angle subtended by diameter = 90° / 直径所对圆周角为90°
Two chords intersect inside a circle / 两弦在圆内相交	Intersecting chords theorem / 相交弦定理

Practice with a variety of problems helps develop the intuition for choosing the correct theorem.

多练习不同类型的题目，有助于培养选择正确定理的直觉。

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Key Theorems in Plane Geometry and Their Applications | 平面几何常用定理及其应用