Circles and Triangles | 圆与三角形

📚 Circles and Triangles | 圆与三角形

In A-Level Mathematics, the interplay between circles and triangles brings together coordinate geometry, trigonometry, and pure geometric theorems. Whether you are finding the equation of a tangent, applying the alternate segment theorem, or calculating the area of a segment, a solid understanding of circles and triangles is essential for Edexcel’s Pure Mathematics papers.

在 A-Level 数学中，圆与三角形的相互关联综合了坐标几何、三角学和纯几何定理。无论是求切线方程、应用弦切角定理还是计算弓形面积，对圆和三角形的扎实理解是应对 Edexcel 纯数试题的关键。

1. Equation of a Circle | 圆的方程

The standard form of a circle centred at (a, b) with radius r is (x – a)² + (y – b)² = r². This is derived directly from the distance formula applied to any point (x, y) on the circumference.

圆心在 (a, b)、半径为 r 的圆的标准方程为 (x – a)² + (y – b)² = r²。该方程直接来源于圆周上任一点 (x, y) 到圆心的距离公式。

If the centre lies at the origin (0,0), the equation simplifies to x² + y² = r². This basic form is often used when analysing intersections with lines or other circles.

若圆心位于原点 (0,0)，方程简化为 x² + y² = r²。在分析直线或圆与圆的交点时，这个基本形式经常使用。

The general expanded form x² + y² + 2gx + 2fy + c = 0 represents a circle with centre (-g, -f) and radius √(g² + f² – c), provided g² + f² – c > 0.

一般展开式 x² + y² + 2gx + 2fy + c = 0 表示圆心为 (-g, -f)、半径为 √(g² + f² – c) 的圆，前提是 g² + f² – c > 0。

2. Finding the Centre and Radius by Completing the Square | 配方法求圆心与半径

To recover the standard form from the general equation, group x-terms and y-terms, then complete the square separately for each variable. For example, x² + 6x + y² – 4y – 3 = 0 becomes (x + 3)² + (y – 2)² = 16, revealing centre (-3, 2) and radius 4.

要从一般式还原为标准式，先将 x 项和 y 项分别归组，再各自配方。例如 x² + 6x + y² – 4y – 3 = 0 化为 (x + 3)² + (y – 2)² = 16，可得圆心 (-3, 2)、半径 4。

Always check that the right-hand side is positive; a negative value indicates no real circle, while zero corresponds to a point circle at the centre. This technique is fundamental for solving intersection and tangency problems.

务必检查等式右边为正；若为负则表示不存在实圆，若为零则对应一个点圆（圆心本身）。这一配方法技巧是解决交点与相切问题的基础。

3. Intersection of a Line and a Circle | 直线与圆的交点

To find the intersection points between a line y = mx + c and a circle, substitute the linear equation into the circle’s equation. This yields a quadratic in x (or y) whose discriminant determines the number of intersection points.

求直线 y = mx + c 与圆的交点时，将直线方程代入圆的方程，得到一个关于 x（或 y）的二次方程，其判别式决定交点个数。

If the discriminant Δ > 0, the line meets the circle at two distinct points. If Δ = 0, the line is tangent to the circle, touching at exactly one point. If Δ < 0, the line and circle do not intersect.

若判别式 Δ > 0，直线与圆交于两个不同点；若 Δ = 0，直线与圆相切，只有一个接触点；若 Δ < 0，直线与圆没有交点。

4. Tangents to a Circle | 圆的切线

A tangent to a circle is a line that touches the circle at exactly one point and is perpendicular to the radius at the point of contact. This perpendicularity provides two powerful methods: using gradients or applying the perpendicular distance formula from the centre to the line.

圆的切线是恰好接触圆于一点的直线，且在该切点处垂直于半径。这种垂直关系提供了两种强有力的方法：利用斜率或应用圆心到直线的垂直距离公式。

For a given point outside the circle, there are generally two tangents. The condition for a line y = mx + c to be tangent to x² + y² = r² is c² = r²(1 + m²), derived from setting the discriminant to zero.

对于圆外给定一点，通常有两条切线。直线 y = mx + c 与圆 x² + y² = r² 相切的条件为 c² = r²(1 + m²)，这来源于令判别式为零。

Alternatively, the distance from the centre (a, b) to the line equals the radius: |Ax₀ + By₀ + C| / √(A² + B²) = r, where the line is Ax + By + C = 0. This approach often simplifies finding unknown coefficients.

另一种方法是圆心 (a, b) 到直线的距离等于半径：|Ax₀ + By₀ + C| / √(A² + B²) = r，其中直线为 Ax + By + C = 0。此方法常能简化未知系数的求解。

5. Chord Properties | 弦的性质

A chord is a line segment joining two points on a circle. The perpendicular from the centre of a circle to a chord bisects the chord. Conversely, the line joining the centre to the midpoint of a chord is perpendicular to the chord.

弦是连接圆上两点的线段。从圆心到弦的垂线平分该弦。反之，连接圆心与弦中点的直线垂直于该弦。

To calculate the length of a chord given the radius r and the perpendicular distance d from the centre to the chord, use the formula chord length = 2√(r² – d²). This is a direct application of Pythagoras’ theorem.

已知半径 r 及圆心到弦的垂距 d，弦长可用公式弦长 = 2√(r² – d²) 计算。这是勾股定理的直接应用。

In coordinate geometry, the chord of intersection of a line and a circle can be found by solving the equations simultaneously and then computing the distance between the two intersection points.

在坐标几何中，直线与圆的交点弦可通过联立方程求解，然后计算两交点之间的距离来得到。

6. Triangles Inscribed in a Circle | 圆内接三角形

A triangle with all vertices on a circle is called an inscribed triangle. One of the most important results is that the angle subtended by a diameter at the circumference is a right angle (90°). This is often called ‘the angle in a semicircle’.

顶点都在圆上的三角形称为圆内接三角形。最重要的结论之一是：直径所对的圆周角为直角（90°）。这常被称为“半圆上的圆周角”。

Using the extended law of sines, for any triangle ABC inscribed in a circle of radius R, a / sin A = b / sin B = c / sin C = 2R. This relationship links side lengths, angles, and the circumradius.

利用扩展的正弦定理，对于任意内接于半径为 R 的圆的三角形 ABC，有 a / sin A = b / sin B = c / sin C = 2R。该关系连接了边长、角度和外接圆半径。

This extended sine rule is invaluable when solving geometric problems involving a circumcircle or when determining the radius of the circle passing through three points.

在处理涉及外接圆的几何问题或确定经过三点的圆的半径时，这个扩展正弦定理不可或缺。

7. The Alternate Segment Theorem | 弦切角定理

The alternate segment theorem states that the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. In other words, ∠(tangent & chord) = angle subtended by the chord in the opposite arc.

弦切角定理指出：切线与经过切点的弦之间的夹角等于该弦所对的另一弧上的圆周角。也就是 ∠(切线-弦) = 弦在另一侧圆弧上所对的圆周角。

This theorem frequently appears in circle geometry proof questions, often combined with angle properties such as the angle at the centre being twice the angle at the circumference.

该定理常出现在圆几何的证明题中，常与圆心角等于两倍圆周角等角度性质联合使用。

Identifying the alternate segment correctly requires careful attention: the chord divides the circle into two segments; the angle in the alternate segment is the one not containing the tangent-chord angle.

正确识别弦切角需要仔细：弦把圆分成两个弓形，另一弓形中的圆周角即为弦切角对应的角。

8. Circumcircle and Circumcentre of a Triangle | 三角形的外接圆与外心

The circle passing through all three vertices of a triangle is its circumcircle. Its centre, the circumcentre, is the intersection point of the three perpendicular bisectors of the sides. The circumcentre is equidistant from all three vertices.

通过三角形三个顶点的圆是其外接圆。外接圆的圆心——外心，是三条边垂直平分线的交点。外心到三个顶点的距离相等。

In coordinate geometry, to find the circumcircle of a triangle with given vertices, you can set up perpendicular bisector equations or solve a system using the general circle equation by substituting the three points. The circumradius R can then be calculated using the distance formula or using R = a / (2 sin A).

在坐标几何中，求给定顶点三角形的外接圆时，可以建立垂直平分线方程，或利用一般式圆方程代入三点求解。外接圆半径 R 可通过距离公式或用 R = a / (2 sin A) 求得。

The circumcentre can lie inside or outside the triangle, depending on whether the triangle is acute, right, or obtuse. For a right-angled triangle, the circumcentre is the midpoint of the hypotenuse.

外心可能在三角形内部、边上或外部，取决于三角形是锐角、直角还是钝角。对于直角三角形，外心是斜边的中点。

9. Radians, Arc Length and Sector Area | 弧度、弧长与扇形面积

Radians are the natural measure for angles in circle geometry. One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. Conversion: π radians = 180°.

弧度是圆几何中角度的自然度量单位。1 弧度是指长度等于半径的弧所对的圆心角。换算关系为：π 弧度 = 180°。

The arc length s of a sector with radius r and angle θ in radians is s = rθ. The area of the sector is A_sector = ½ r² θ. These formulas are essential for calculating lengths and areas within circles.

半径为 r、圆心角为 θ 弧度的扇形，其弧长 s = rθ，扇形面积 A_sector = ½ r² θ。这些公式对于圆内长度和面积的计算至关重要。

When the angle is given in degrees, first convert to radians by multiplying by π/180. Using radians avoids awkward conversions and makes calculus with trigonometric functions much simpler.

当角度以度给出时，先乘以 π/180 转换为弧度。使用弧度可以避免繁琐的换算，并使得三角函数的微积分运算更简洁。

10. Area of a Triangle and Segment Area | 三角形面积与弓形面积

The area of a triangle can be expressed as ½ ab sin C, where a and b are two sides and C is the included angle. This formula is particularly useful when the triangle forms part of a sector or a segment in a circle.

三角形面积可表示为 ½ ab sin C，其中 a 和 b 为两边，C 为夹角。当三角形作为扇形或弓形的一部分时，该公式尤为有用。

A segment of a circle is the region bounded by a chord and the arc. The area of a segment (for θ in radians) is A_segment = ½ r² (θ – sin θ). This is derived by subtracting the area of the isosceles triangle from the area of the sector.

弓形是由弦与弧围成的区域。弓形面积（θ 以弧度计）为 A_segment = ½ r² (θ – sin θ)。它是由扇形面积减去等腰三角形面积推导而来的。

In many problems, you will need to find the area of the triangular part using ½ r² sin θ for the two radii and included angle, then combine with sector area to find shaded regions.

在许多问题中，你需要利用 ½ r² sin θ 计算由两条半径和夹角构成的三角形面积，然后结合扇形面积求解阴影部分。

11. Using the Cosine Rule with Circles | 余弦定理在圆中的应用

When a triangle inside a circle has known side lengths or angles, the cosine rule a² = b² + c² – 2bc cos A helps determine missing lengths or angles. This is especially useful when the triangle is not right-angled and the sine rule alone is insufficient.

当圆内三角形已知边长或角度时，余弦定理 a² = b² + c² – 2bc cos A 可用于求解缺失的边长或角度。当三角形不是直角三角形而正弦定理又不够用时，余弦定理尤其有用。

Combined with the extended sine rule, you can find the circumradius R without constructing perpendicular bisectors: R = a / (2 sin A) = abc / (4 × area of triangle).

结合扩展正弦定理，你无需作垂直平分线即可求出外接圆半径 R：R = a / (2 sin A) = abc / (4 × 三角形面积)。

Such problems often appear in examinations where you must compute an angle, then use it to find an arc length or the area of a related segment, effectively linking triangles and circles.

这类问题常出现在考试中，你需要先求出一个角，然后用它计算弧长或相关弓形的面积，有效连接了三角形与圆的知识。

12. Common Exam Pitfalls and Summary | 常见考试陷阱与总结

Common mistakes include forgetting to convert degrees to radians when using sector formulas, misapplying the tangent condition by using gradient alone instead of the perpendicular distance, and misidentifying the alternate segment. Always double-check the sign when completing the square for the general circle equation.

常见的错误包括：在使用扇形公式时忘记将度转换为弧度；错误应用切线条件，仅用斜率而忽略了垂直距离；以及错误识别弦切角中的另一弓形。在用一般式配方时务必复核符号。

Mastering the connection between triangles and circles means you can confidently solve problems involving coordinate geometry, trigonometry, and geometric proofs. Regular practice with past Edexcel questions, paying attention to these links, will build the fluency required for high marks.

掌握三角形与圆的联系意味着你将能自信地解决涉及坐标几何、三角学和几何证明的问题。通过经常练习 Edexcel 历年真题并关注这些关联，将培养出获取高分所需的流畅度。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Circles and Triangles | 圆与三角形