📚 Intersecting Chords Theorems | 相交弦定理
The intersecting chords theorems are a cornerstone of circle geometry in the Edexcel IGCSE Mathematics syllabus. These elegant results tell us how the lengths of segments formed by intersecting chords, secants, and tangents relate to one another. Mastering these theorems is essential for solving a wide range of circle geometry problems, from straightforward length calculations to multi-step reasoning questions.
相交弦定理是 Edexcel IGCSE 数学课程中圆几何的核心内容。这些精妙的结论揭示了相交弦、割线与切线所形成的线段长度之间的数量关系。掌握这些定理对于解决各类圆几何问题至关重要，无论是直接计算长度，还是需要多步推理的综合题型。
1. Key Terminology | 关键术语
Before exploring the theorems, we must first clarify the precise meaning of three central terms. A chord is a straight line segment with both endpoints lying on the circumference of the circle. A secant is a straight line that intersects the circle at two distinct points, effectively extending a chord beyond the circle. A tangent is a straight line that touches the circle at exactly one point, called the point of tangency.
在深入定理之前，我们必须首先明确三个核心术语的准确含义。弦是两端点均在圆周上的线段。割线是与圆相交于两个不同点的直线，可视为弦向圆外的延伸。切线是恰好与圆只有一个公共点的直线，该公共点称为切点。
When a chord, secant, or tangent line is drawn from a common point, the distances from that point to the circle along each line are called the segment lengths. The theorems we study next relate these segment lengths through multiplication.
当弦、割线或切线从同一点引出时，从该点到圆沿各直线的距离称为线段长度。接下来要研究的定理正是通过乘法运算将这些线段长度联系起来。
2. The Intersecting Chords Theorem (Interior Case) | 相交弦定理（圆内情形）
The most fundamental version of the theorem applies when two chords intersect inside a circle. Consider a circle containing two chords AB and CD that intersect at point P in the interior of the circle. The theorem states that the product of the segments of one chord equals the product of the segments of the other chord:
该定理最基本的形式适用于两条弦在圆内相交的情形。设圆内有两条弦 AB 和 CD，它们在圆内一点 P 相交。定理指出：一条弦被分成的两段长度之积等于另一条弦两段长度之积：
PA × PB = PC × PD
In plain words, if you multiply the two shorter distances from P to the endpoints along one chord, you obtain exactly the same value as doing the same for the other chord. This relationship holds for any pair of intersecting chords, provided the intersection point lies inside the circle.
通俗地说，沿着一条弦，将 P 到两个端点的距离相乘，得到的数值与对另一条弦做同样的乘法所得数值完全相同。只要交点位于圆内，这一关系对任意一对相交弦均成立。
3. Proof of the Interior Theorem | 圆内相交弦定理的证明
To prove PA × PB = PC × PD, we begin by constructing two auxiliary lines: join A to D, and join B to C. This creates two triangles, ΔAPD and ΔCPB, inside the circle. Our goal is to show that these two triangles are similar.
为了证明 PA × PB = PC × PD，我们首先构造两条辅助线：连接 A 与 D，连接 B 与 C。这就在圆内构造了两个三角形：ΔAPD 和 ΔCPB。我们的目标是证明这两个三角形相似。
First, ∠APD = ∠CPB because they are vertically opposite angles formed by the intersection of lines AB and CD. Second, ∠DAP = ∠BCP because both angles subtend the same chord BD — they are angles in the same segment of the circle. Since two pairs of corresponding angles are equal, ΔAPD is similar to ΔCPB by the AA (Angle-Angle) criterion.
首先，因为 AB 和 CD 是两条相交直线，对顶角相等，所以 ∠APD = ∠CPB。其次，∠DAP = ∠BCP，因为这两个角都对应同一条弧 BD——它们是圆中同弧所对的圆周角。由于两对对应角相等，根据 AA（角-角）判定准则，ΔAPD 与 ΔCPB 相似。
From the similarity, the corresponding sides are in proportion: AP/CP = PD/PB. Cross-multiplying yields AP × PB = CP × PD, which is precisely the intersecting chords theorem. This proof demonstrates that the theorem is a natural consequence of similarity, a foundation concept you already know well.
由相似性可知对应边成比例：AP/CP = PD/PB。交叉相乘得到 AP × PB = CP × PD，这正是相交弦定理的结论。该证明说明此定理是相似三角形这一基础概念的必然推论。
4. The Intersecting Secants Theorem (Exterior Case) | 相交割线定理（圆外情形）
When the intersection point lies outside the circle, a closely related theorem applies. Suppose two secants are drawn from an external point P, with the first secant intersecting the circle at points A (farther) and B (nearer), and the second secant intersecting at points C (farther) and D (nearer). Then the same product relationship holds:
当交点位于圆外时，一个紧密相关的定理依然适用。假设从圆外一点 P 引两条割线，第一条割线与圆交于点 A（较远）和 B（较近），第二条割线与圆交于点 C（较远）和 D（较近）。此时同样的乘积关系仍然成立：
PA × PB = PC × PD
It is crucial to note that PB and PD are the nearer intersection distances from P, while PA and PC are the farther ones. A common error is to substitute the chord length AB directly into the formula. Always multiply the full distance from P to the far intersection by the distance from P to the near intersection.
必须注意，PB 和 PD 是 P 到较近交点的距离，而 PA 和 PC 是 P 到较远交点的距离。一个常见错误是将弦的长度 AB 直接代入公式。务必用 P 到远交点的距离乘以 P 到近交点的距离。
This theorem can be proved similarly by joining appropriate points on the circle to create similar triangles, using the cyclic quadrilateral exterior-angle property in place of the same-segment angle property used in the interior case.
该定理的证明方法类似，只需连接圆上适当的两点来构造相似三角形，并用圆内接四边形外角等于内对角的性质替代内情形中使用的同弧所对圆周角相等性质即可。
5. The Secant-Tangent Theorem | 切割线定理
The third variation combines a tangent and a secant. If a tangent PT touches the circle at T and a secant PAB intersects the circle at points A and B from the same external point P, then the square of the tangent length equals the product of the secant’s segments:
第三种变体将切线与割线结合在一起。如果从同一点 P 引一条切线 PT 切圆于点 T，再引一条割线 PAB 与圆交于点 A 和 B，则切线长度的平方等于割线两段长度之积：
PT² = PA × PB
Here, A is the farther intersection and B is the nearer intersection along the secant. The theorem is elegantly consistent with the previous forms: we can think of the tangent as a “degenerate secant” where the two intersection points coincide at T, so PA and PB both collapse to PT.
其中，A 是割线上的远交点，B 是近交点。此定理与前两种形式有着优美的统一性：我们可以将切线视为一条”退化了的割线”，其两个交点重合于点 T，于是 PA 和 PB 都退化为 PT。
The proof again relies on similarity: joining T to A and B yields ΔPTA and ΔPBT. One can show ∠PTA = ∠PBT (tangent-chord theorem) and ∠APT = ∠TPB (common angle), establishing similarity and leading to PA/PT = PT/PB, from which PT² = PA × PB follows directly.
该定理的证明同样依赖于相似三角形：连接 T 与 A、T 与 B 可得到 ΔPTA 和 ΔPBT。由弦切角定理可知 ∠PTA = ∠PBT，又 ∠APT = ∠TPB（公共角），从而两三角形相似，得到 PA/PT = PT/PB，据此直接推出 PT² = PA × PB。
6. Unifying the Three Theorems | 三个定理的统一
Although the three theorems appear at first glance to be separate results, they are in fact one unified principle. In every case, take a point P and two lines through P, each meeting the circle at two points (in the tangent case, the two points coincide). The product of the distances from P to the two intersection points along the first line equals the product along the second line.
虽然这三个定理乍看之下似乎是各自独立的结论，但事实上它们统一于同一条原理。在每一种情形中，取一点 P 和经过 P 的两条直线，每条直线与圆交于两个点（切线情形中两点重合）。沿第一条直线从 P 到两个交点的距离之积，等于沿第二条直线从 P 到两个交点的距离之积。
This unified view is powerful: it means that once you understand one form, you can easily recall the others by observing whether the intersection point is inside the circle (chords), outside the circle (secants), or whether one line is a tangent. The formula structure remains identical throughout.
这种统一的视角非常有用：一旦理解了其中一种形式，只需观察交点是位于圆内（弦）、圆外（割线），还是其中一条线为切线，即可轻松想起其他两种形式。无论哪种情况，公式的核心结构始终不变。
On Edexcel IGCSE papers, the exterior and tangent-secant forms are frequently tested in calculations of unknown lengths, so familiarity with all three is essential for full marks on circle theorem questions.
在 Edexcel IGCSE 考试中，圆外割线和切割线这两种形式经常用于未知长度的计算，因此熟练掌握全部三种形式是圆定理题目得满分的关键。
7. Worked Example 1: Interior Chords | 例题一：圆内相交弦
Problem. Two chords AB and CD intersect at point P inside a circle. Given PA = 6 cm, PB = 4 cm, and PC = 8 cm, calculate the length of PD.
题目：两条弦 AB 和 CD 在圆内一点 P 处相交。已知 PA = 6 cm，PB = 4 cm，PC = 8 cm，求 PD 的长度。
Solution. Apply the intersecting chords theorem directly:
解答：直接运用相交弦定理：
PA × PB = PC × PD
6 × 4 = 8 × PD
PD = 24 ÷ 8 = 3 cm
The required length PD is 3 cm. Notice how quickly the answer emerges — the theorem converts a potentially complicated geometric situation into a simple linear equation.
所求 PD 的长度为 3 cm。可以看到答案迅速得出——该定理将一个原本可能复杂的几何情境转化为一个简单的一元一次方程。
8. Worked Example 2: Exterior Secants | 例题二：圆外割线
Problem. From an external point P
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