Alternate Segment Theorem | 弦切角定理

📚 Alternate Segment Theorem | 弦切角定理

The Alternate Segment Theorem, also known as the Tangent-Chord Theorem, is one of the most powerful and frequently tested circle theorems in Edexcel IGCSE Mathematics. It establishes a beautifully simple relationship between a tangent, a chord, and the angles they create within a circle. Once mastered, this theorem transforms seemingly complex angle-chasing problems into straightforward applications of a single rule.

弦切角定理，又称切线-弦定理，是 Edexcel IGCSE 数学中最重要且高频考查的圆定理之一。它揭示了一条切线、一条弦以及它们在圆内所形成角度之间极其简洁而优美的关系。一旦掌握了这一定理，许多看似复杂的角度计算问题都能转化为对一条规则的直接应用。

1. What is the Alternate Segment Theorem? | 什么是弦切角定理？

The theorem states: The angle between a tangent and a chord drawn through the point of contact is equal to the angle in the alternate segment.

定理内容：经过切点的切线与弦所夹的角，等于该弦所对的、位于切线另一侧的圆周角（即交替弓形中的角）。

To understand this definition, imagine a circle with a tangent line touching it at a single point, say point A. From A, draw a chord AB. This chord divides the circle into two regions called segments. The angle between the tangent line and chord AB equals the angle subtended by AB at any point on the arc opposite to the tangent.

为了理解这一定义，假设一个圆有一条切线在点 A 与圆相切。从 A 作一条弦 AB，这条弦将圆分成两个区域，称为弓形。切线与弦 AB 的夹角，等于 AB 在位于切线对侧圆弧上任意一点所张的角度。

This theorem is fundamental for solving angle problems in circle geometry, and Edexcel IGCSE exams regularly include questions that require its application.

这一定理是解决圆几何角度问题的基石，Edexcel IGCSE 考试中经常出现需要运用该定理的题目。

2. Key Terminology | 关键术语

Before applying the theorem, you must be comfortable with four essential terms:

在应用该定理之前，你需要熟练掌握四个关键术语：

Tangent: a straight line that touches the circle at exactly one point, called the point of contact.
Chord: a straight line segment whose two endpoints lie on the circumference of the circle.
Segment: the region of the circle bounded by a chord and the arc it cuts off.
Alternate segment: the segment lying on the opposite side of the chord from the tangent; it is the region that does not contain the tangent line.

切线：与圆恰好只有一个公共点的直线，该点称为切点。
弦：两个端点都在圆周上的线段。
弓形：由一条弦与其所截的圆弧围成的区域。
交替弓形：相对于切线位于弦的另一侧的弓形，即不包含切线的那个区域。

In Edexcel IGCSE papers, you will often see the tangent labelled with two letters, such as XY, where X and Y are points on the tangent line, and the point of contact is labelled separately.

在 Edexcel IGCSE 试卷中，切线通常用两个字母标注，例如 XY，其中 X 和 Y 是切线上的两个点，而切点会单独标注。

3. Understanding the Alternate Segment | 理解”交替弓形”

The phrase “alternate segment” confuses many students, so let us examine it carefully. Consider a circle with tangent XY touching at point A, and chord AB drawn from A. The chord AB divides the circle into two segments: one is on the same side of AB as the tangent, and the other is on the opposite side.

“交替弓形”这个概念让许多学生感到困惑，我们仔细来分析。设圆在点 A 处有切线 XY，并从 A 作弦 AB。弦 AB 将圆分成两个弓形：一个与切线在 AB 的同侧，另一个在 AB 的对侧。

The alternate segment is the one on the opposite side of chord AB from the tangent line. If the tangent lies above the chord, the alternate segment is the lower segment of the circle, and vice versa.

交替弓形就是与切线分居弦 AB 两侧的那个弓形。如果切线在弦的上方，那么交替弓形就是圆的较下方的弓形区域，反之亦然。

It is crucial to note that there are two angles between the tangent and the chord: one acute and one obtuse, since the tangent line extends in both directions. Each of these angles corresponds to one of the two segments. The acute angle between the tangent and the chord equals the angle in the segment on one side, while the obtuse angle equals the angle in the segment on the other side.

特别注意：切线与弦之间存在两个夹角——一个锐角和一个钝角，因为切线向两个方向延伸。每个角分别对应其中一个弓形。切线与弦的锐角等于一侧弓形中的圆周角，而钝角等于另一侧弓形中的圆周角。

∠BAY = ∠ACB and ∠BAX = ∠ADB

In this notation, Y is a point on the tangent on one side of A, X is a point on the tangent on the other side of A, C lies on one arc of AB, and D lies on the other arc.

在此符号中，Y 是切线上位于 A 一侧的点，X 是切线上位于 A 另一侧的点，C 在 AB 的其中一条弧上，D 在另一条弧上。

4. The Theorem in Symbolic Form | 定理的符号表达

Let us set up a standard notation that appears in most Edexcel IGCSE problems. Suppose line XY is tangent to the circle at point A, and AB is a chord. Let C be any point on the major arc AB (the longer arc), and let D be any point on the minor arc AB (the shorter arc).

我们建立一个 Edexcel IGCSE 题目中常见的标准记号。设直线 XY 在点 A 与圆相切，AB 是弦。设 C 是优弧 AB（较长弧）上任意一点，D 是劣弧 AB（较短弧）上任意一点。

The theorem gives us two equalities:

该定理给出两个等式：

∠BAY = ∠ACB

∠BAX = ∠ADB

Where ∠BAY is the angle between chord AB and tangent AY, and ∠ACB is the angle subtended by chord AB at point C on the alternate arc.

其中 ∠BAY 是弦 AB 与切线 AY 之间的夹角，∠ACB 是弦 AB 在交替弧上点 C 处所张的圆周角。

Notice that as C moves along the same arc, the value of ∠ACB remains constant because angles in the same segment are equal. This consistency makes the Alternate Segment Theorem extremely reliable for angle chasing.

注意，当 C 在同一条弧上移动时，∠ACB 的值保持不变，因为同弓形内的圆周角相等。这种一致性使得弦切角定理在角度推导中极为可靠。

5. Why It Works: A Proof Sketch | 为什么成立：证明思路

Understanding the proof not only satisfies curiosity but also helps you remember the theorem and see how circle theorems interconnect. The proof uses two other fundamental circle theorems.

理解证明过程不仅能满足好奇心，还能帮助你记住定理并体会各圆定理之间的内在联系。该证明用到两条基本的圆定理。

Step 1: Let O be the centre of the circle. Since OA is a radius and XY is a tangent at A, the radius-tangent property states that OA is perpendicular to XY. Therefore ∠OAY = 90°.

第一步：设 O 为圆心。因为 OA 是半径，XY 是过 A 的切线，根据半径-切线性质，OA 垂直于 XY，故 ∠OAY = 90°。

Step 2: The angle ∠BAY = 90° − ∠OAB. In triangle OAB, since OA = OB (both are radii), the triangle is isosceles, so ∠OBA = ∠OAB. The angle sum of a triangle gives ∠AOB = 180° − 2∠OAB.

第二步：∠BAY = 90° − ∠OAB。在三角形 OAB 中，因为 OA = OB（均为半径），该三角形为等腰三角形，故 ∠OBA = ∠OAB。由三角形内角和为 180° 可得 ∠AOB = 180° − 2∠OAB。

Step 3: Now consider ∠ACB. The angle at the centre theorem states that ∠AOB = 2∠ACB, because ∠AOB is the angle at the centre subtended by the same chord AB.

第三步：再考虑 ∠ACB。根据圆心角定理，∠AOB = 2∠ACB，因为 ∠AOB 是同一条弦 AB 所对的圆心角。

Step 4: Combining these results, we have 2∠ACB = 180° − 2∠OAB. Dividing by 2 gives ∠ACB = 90° − ∠OAB. But from Step 2, ∠BAY = 90° − ∠OAB. Therefore ∠BAY = ∠ACB. The theorem is proved.

第四步：综合以上结果，2∠ACB = 180° − 2∠OAB。两边除以 2 得 ∠ACB = 90° − ∠OAB。由第二步知 ∠BAY = 90° − ∠OAB，因此 ∠BAY = ∠ACB。定理得证。

6. Basic Application: Finding Angles | 基础应用：求角度

Let us work through a simple example to see the theorem in action.

我们通过一个简单例题来感受定理的实际运用。

Example: In the diagram below, line XY is tangent to the circle at point A. Chord AB is drawn, and C is a point on the major arc AB. Given that ∠BAY = 48°, find ∠ACB.

例题：如下图所示，直线 XY 在点 A 与圆相切，作弦 AB，C 是优弧 AB 上一点。已知 ∠BAY = 48°，求 ∠ACB。

Solution: By the Alternate Segment Theorem, the angle between the tangent AY and chord AB equals the angle in the alternate segment at C. Hence ∠ACB = ∠BAY = 48°.

解答：根据弦切角定理，切线 AY 与弦 AB 的夹角等于交替弓形中点 C 所对应的圆周角。因此 ∠ACB = ∠BAY = 48°。

Now suppose we are told that ∠ACB = 63° and asked to find ∠BAX (the angle on the other side of the tangent). Since ∠BAX is the angle between tangent AX and chord AB, it equals the angle in the segment opposite to it. The two angles ∠BAX and ∠BAY lie on a straight line, so ∠BAX = 180° − 63° = 117°. This matches the obtuse angle in the other segment.

现在假设已知 ∠ACB = 63°，要求 ∠BAX（切线另一侧的夹角）。由于 ∠BAX 是切线 AX 与弦 AB 的夹角，它等于其对侧弓形中的角。∠BAX 与 ∠BAY 在同一条直线上互补，故 ∠BAX = 180° − 63° = 117°。这与另一弓形中的钝角一致。

7. Combining with Isosceles Triangles | 与等腰三角形的结合运用

Many Edexcel IGCSE questions combine the Alternate Segment Theorem with the properties of isosceles triangles. This is especially common when the chord is a radius or when two sides of a triangle within the circle are equal.

许多 Edexcel IGCSE 题目将弦切角定理与等腰三角形的性质结合考查。当弦恰为半径或圆内三角形有两边相等时，这种结合尤为常见。

Example: A circle has centre O. The tangent at point T on the circle is drawn, and a chord TP is drawn such that OT = TP. Find the angle between the tangent and TP given that ∠OTP = 70°.

例题：圆以 O 为圆心。过圆上点 T 作切线，并作弦 TP，使 OT = TP。已知 ∠OTP = 70°，求切线与弦 TP 的夹角。

Solution: Since OT = TP, triangle OTP is isosceles with equal sides OT and TP. The base angles at O and P are equal. Since ∠OTP = 70°, the base angles are each (180° − 70°) ÷ 2 = 55°.

解答：因为 OT = TP，三角形 OTP 是等腰三角形，两腰 OT 与 TP 相等。底角 ∠TOP 与 ∠TPO 相等。由于 ∠OTP = 70°，每个底角为 (180° − 70°) ÷ 2 = 55°。

Now consider the tangent at T. The radius OT is perpendicular to the tangent. The angle between the tangent and chord TP is 90° − ∠OTP = 90° − 70° = 20°. Alternatively, by the Alternate Segment Theorem, this angle equals the angle in the alternate segment, which is ∠TOP = 55°? Let us check carefully.

现在考虑过 T 的切线。半径 OT 垂直于切线。切线与弦 TP 的夹角为 90° − ∠OTP = 90° −

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Alternate Segment Theorem | 弦切角定理