Power of a Point and Intersecting Chords | 圆幂定理与相交弦应用

📚 Power of a Point and Intersecting Chords | 圆幂定理与相交弦应用

The power of a point theorem is a fundamental result in Euclidean geometry that unifies several chord-related theorems. It states that for a point P and a circle, the product of the distances from P to the two intersection points of any secant line through P with the circle is constant.

圆幂定理是欧几里得几何中的一个基本结论，它统一了多个与弦相关的重要定理。该定理指出：对于平面内一点 P 和一个圆，过 P 的任意一条割线与圆交于两点，则 P 到这两个交点距离的乘积恒为定值。

1. Statement of the Power of a Point | 圆幂定理的表述

Let P be a point in the plane and let a line through P intersect a circle at points A and B. Then the value PA × PB is independent of the chosen line. This constant is called the power of the point P with respect to the circle.

设 P 为平面内一点，过 P 的直线与圆交于 A、B 两点，则乘积 PA × PB 与所选直线的位置无关。这个定值称为点 P 关于该圆的幂。

Power(P) = PA × PB = |d² − r²|

Here d is the distance from P to the center O of the circle, and r is the radius. If P lies outside the circle the power is positive; if P lies inside the circle the power is negative; if P lies on the circle the power is zero.

其中 d 是点 P 到圆心 O 的距离，r 是半径。若 P 在圆外，则幂为正；若 P 在圆内，则幂为负；若 P 在圆上，则幂为零。

2. The Intersecting Chords Theorem | 相交弦定理

When point P lies inside a circle, draw two chords AB and CD that intersect at P. The theorem states that PA × PB = PC × PD.

当点 P 位于圆内时，作两条相交于 P 的弦 AB 和 CD，则相交弦定理指出：PA × PB = PC × PD。

PA × PB = PC × PD

This is the most direct application of the power of a point. Notice that P divides each chord into two segments, and the products of the two segment lengths on each chord are equal.

这是圆幂定理最直接的应用。注意 P 将每条弦分为两段，而每条弦上两段长度的乘积相等。

3. The Secant-Secant Theorem | 割线定理

When P lies outside the circle, consider two secant lines through P. The first secant meets the circle at points A and B, with A closer to P. The second secant meets the circle at points C and D, with C closer to P. Then PA × PB = PC × PD.

当点 P 在圆外时，考虑过 P 的两条割线。第一条割线与圆交于 A、B 两点，且 A 离 P 较近；第二条割线与圆交于 C、D 两点，且 C 离 P 较近。则割线定理指出：PA × PB = PC × PD。

PA × PB = PC × PD

Here the entire secant segment from P to the farther intersection point is used. This theorem is useful for finding unknown distances when two external secants are drawn.

这里使用的是从 P 到较远交点的整条割线线段。该定理在已知两条外割线时求未知距离非常有用。

4. The Tangent-Secant Theorem | 切割线定理

If P lies outside the circle, and a tangent from P touches the circle at point T, while a secant from P meets the circle at points A and B, then the square of the tangent length equals the product of the secant segments: PT² = PA × PB.

若点 P 在圆外，从 P 引圆的切线切圆于点 T，同时从 P 作割线交圆于 A、B 两点，则切线长的平方等于割线两段之积：PT² = PA × PB。

PT² = PA × PB

This can be viewed as the limiting case of the secant-secant theorem where the two intersection points C and D of the second secant coalesce into the single tangent point T.

这可以看作割线定理的极限情形：当第二条割线的两个交点 C、D 逐渐重合为切点 T 时，便得到切割线定理。

5. Unification of the Three Theorems | 三个定理的统一

The intersecting chords, secant-secant, and tangent-secant theorems are all special cases of the power of a point. The sign convention in the algebraic definition automatically handles the interior and exterior cases.

相交弦定理、割线定理和切割线定理都是圆幂定理的特殊情形。代数定义中的符号约定可以自动处理圆内和圆外两种情形。

PA × PB = PC × PD = constant

For an interior point, the signed power is negative because the two directed segments have opposite directions. For an exterior point, the power is positive.

对于圆内一点，由于两条有向线段方向相反，幂取负值；对于圆外一点，幂为正值。

6. Example: Finding a Chord Segment | 例题：求弦的线段长度

In a circle, two chords AB and CD intersect at P. Given PA = 4, PB = 6, and PC = 3, find PD.

在圆中，两条弦 AB 与 CD 相交于点 P。已知 PA = 4，PB = 6，PC = 3，求 PD。

By the intersecting chords theorem, PA × PB = PC × PD. Substituting the values gives 4 × 6 = 3 × PD, so PD = 8.

由相交弦定理，PA × PB = PC × PD。代入数值得 4 × 6 = 3 × PD，解得 PD = 8。

4 × 6 = 3 × PD ⇒ PD = 8

This demonstrates how the theorem reduces a geometric problem to a simple algebraic equation.

这个例子说明，利用圆幂定理可以将几何问题简化为简单的代数方程。

7. Example: Tangent Length and Secant | 例题：切线与割线的计算

From a point P outside a circle, a tangent PT has length 5, and a secant through P intersects the circle at A and B with PA = 4. Find PB.

在圆外一点 P，切线 PT 的长为 5，过 P 的割线交圆于 A、B，且 PA = 4，求 PB。

By the tangent-secant theorem, PT² = PA × PB, so 25 = 4 × PB, giving PB = 25/4 = 6.25.

由切割线定理，PT² = PA × PB，所以 25 = 4 × PB，得 PB = 25/4 = 6.25。

PT² = PA × PB ⇒ 5² = 4 × PB ⇒ PB = 25/4

Notice that the entire external secant segment PB includes both PA and AB. Therefore AB = PB − PA = 6.25 − 4 = 2.25.

注意整个外部割线线段 PB 包含 PA 和 AB 两部分，因此 AB = PB − PA = 6.25 − 4 = 2.25。

8. Connection with Similar Triangles | 与相似三角形的联系

The power of a point theorem can be proved using similar triangles. For example, in the intersecting chords case, triangles APD and CPB are similar because their corresponding angles are equal.

圆幂定理可以用相似三角形来证明。例如，在相交弦的情形中，△APD 与 △CPB 相似，因为它们的对应角相等。

∠APD = ∠CPB, ∠ADP = ∠CBP

From this similarity we obtain PA/PC = PD/PB, which upon cross-multiplication gives PA × PB = PC × PD. This perspective helps students see the theorem as a consequence of proportional sides in similar triangles.

由相似得 PA/PC = PD/PB，交叉相乘后即得 PA × PB = PC × PD。这一视角帮助学生理解圆幂定理是相似三角形对应边成比例的必然结果。

9. Common Problem Types and Tips | 常见题型与技巧

Problems involving the power of a point can be classified into three main types: finding missing lengths, proving equality of products, and establishing that four points are concyclic.

涉及圆幂定理的题目主要可以归为三类：求未知长度、证明乘积相等、证明四点共圆。

When solving, always identify whether P is inside or outside the circle, then choose the correct formula. When a tangent is present, remember that the tangent length appears only once in the product.

解题时，先判断点 P 在圆内还是圆外，再选择正确的公式。若题目中出现切线，注意切线长在乘积中只出现一次。

If the problem involves two chords, a secant and a tangent, or two secants, draw the diagram and label all known segments before applying the theorem.

如果题目涉及两条弦、一条割线与一条切线、或两条割线，先作图并标出所有已知线段，再套用定理。

10. Application in Construction Problems | 在作图题中的应用

The power of a point is also a powerful tool in geometric construction. For example, to construct a tangent of given length from an external point, one can fix the secant segment using the relation PT² = PA × PB.

圆幂定理也是几何作图的有力工具。例如，要从圆外一点作长度为给定值的切线，可利用关系 PT² = PA × PB 来确定割线位置。

Because the product determines the power, one can construct a point on a given line using a circle and a known segment. This technique appears in various advanced geometry competitions.

由于乘积确定了幂的值，可以用一个圆和一条已知线段在给定直线上构造对应点。这一技巧常出现在高级几何竞赛中。

11. Relation to the Radical Axis | 与根轴的关系

For two circles, the locus of points that have equal power with respect to both circles is a straight line called the radical axis. If the circles intersect, the radical axis is the common chord line.

对于两个圆，具有相等幂的点的轨迹是一条直线，称为根轴。若两圆相交，根轴就是两圆的公共弦所在的直线。

Power₁(P) = Power₂(P)

This concept extends the power of a point beyond a single circle and is essential for solving systems of circles. Understanding it deepens the student’s grasp of the fundamental theorem.

这一概念将圆幂定理从单个圆推广到圆系，是解决圆系问题的重要工具。理解它可以加深学生对圆幂定理本质的认识。

12. Summary and Exam Tips | 总结与考试要点

The power of a point unifies intersecting chords, secants, and tangents into one elegant idea. In exams, the direct application of these formulas often appears as part of a larger proof or calculation.

圆幂定理将相交弦、割线和切线统一为一个简洁的结论。在考试中，直接套用这些公式通常作为更复杂证明或计算的一部分出现。

Students should memorize the three main forms, practice converting between them, and always verify whether the point is internal or external. A carefully drawn diagram is half the solution.

学生应熟记三种主要形式，练习它们的相互转化，并始终确认点在圆内还是圆外。画好示意图往往等于解决了一半问题。

By mastering the power of a point, you gain a powerful and flexible tool that simplifies many geometry problems in both IGCSE and A-level examinations.

掌握圆幂定理，你就拥有了一件强大而灵活的几何工具，能够简化 IGCSE 和 A-level 考试中的许多几何问题。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Power of a Point and Intersecting Chords | 圆幂定理与相交弦应用