📚 Constructing Perpendicular Bisectors, Angle Bisectors and Loci | 构造垂直平分线、角平分线与轨迹
Geometric constructions are a fundamental part of KS3 mathematics, allowing you to create precise figures using only a compass and a straightedge. In this article, we will explore how to construct perpendicular bisectors, angle bisectors, and understand loci — sets of points that satisfy certain conditions. Mastering these skills not only helps in solving geometry problems but also builds a deeper understanding of mathematical reasoning and accuracy.
几何构造是KS3数学的基础部分，只需使用圆规和直尺即可精确绘制图形。本文将探讨如何构造垂直平分线、角平分线，并理解轨迹 —— 即满足特定条件的点集。掌握这些技能不仅有助于解决几何问题，还能加深对数学推理和精确性的理解。
1. Introduction to Constructions | 构造简介
In geometry, a construction is a precise drawing made using specific tools and following a set of rules. The two classic tools for constructions are a compass and an unmarked straightedge. The compass is used to draw circles and arcs, and to transfer lengths. The straightedge is used solely for drawing straight lines; it has no measurement markings, so you cannot measure lengths or angles directly with it.
在几何中，构造是使用特定工具并遵循一组规则进行的精确绘图。用于构造的两种经典工具是圆规和无刻度直尺。圆规用于画圆和弧线，以及转移长度。直尺仅用于画直线；它没有测量刻度，因此无法直接用其测量长度或角度。
All construction steps rely on basic geometric properties: circles have a constant radius, the intersection of arcs define points equidistant from given points, and so on. No measuring with a ruler or protractor is allowed in a pure construction; you rely on the relationships between points and lines.
所有构造步骤都依赖于基本的几何性质：圆有固定的半径，弧线的交点定义了与给定点等距的点，等等。在纯构造中，不允许使用直尺或量角器进行测量；你需要依赖于点与线之间的关系。
2. Equipment Needed for Accurate Constructions | 精确构造所需的工具
To perform geometric constructions accurately, you will need a compass with a sharp pencil lead, an unmarked straightedge (often a ruler without markings or by ignoring the markings), a sharp pencil, and an eraser for corrections. Some students find a pair of compasses with a locking mechanism helpful to maintain a fixed radius.
为了准确地执行几何构造，你需要一支带有锋利铅芯的圆规、一把无刻度直尺（通常是一把忽略刻度的尺子）、一支削尖的铅笔和一块用于修改的橡皮。有些学生发现带有锁定机构的圆规有助于保持固定的半径。
Always ensure your compass is firm and the pencil is sharpened to a fine point. Any wobble in the compass can lead to inaccurate arcs and incorrect constructions. Practice drawing arcs and circles without changing the radius to gain control.
始终确保圆规稳固，铅笔削得很尖。圆规的任何晃动都可能导致弧线不准确和构造错误。练习不变半径地画弧线和圆，以获得控制力。
3. Constructing a Perpendicular Bisector of a Line Segment | 构造线段的垂直平分线
The perpendicular bisector of a line segment is a line that divides the segment into two equal lengths at a right angle (90°). Every point on the perpendicular bisector is equidistant from the two endpoints of the segment. To construct it:
线段的垂直平分线是一条将该线段分成两个相等长度且相交成直角（90°）的线。垂直平分线上的每一点到线段两端点的距离相等。构造步骤如下：
Step 1: Place the compass point at one endpoint of the segment (call it A) and open the compass to more than half the length of AB. Draw an arc above and below the segment.
步骤一：将圆规尖端置于线段的一个端点（记为 A），将圆规开口调整为大于 AB 长度的一半。在线段的上方和下方各画一条弧线。
Step 2: Without changing the compass width, move the compass point to the other endpoint B and draw two arcs that intersect the first arcs above and below the segment. Label the intersection points as P and Q.
步骤二：保持圆规宽度不变，将圆规尖端移到另一端点 B，画两条弧线，与先前画出的弧线在线段上、下方相交。将交点标记为 P 和 Q。
Step 3: Use the straightedge to draw a line through points P and Q. This line is the perpendicular bisector of AB. It crosses AB at its midpoint M, and the angle between AB and PQ is 90°.
步骤三：用直尺过点 P 和 Q 画一条直线。这条线就是 AB 的垂直平分线。它与 AB 相交于中点 M，并且 AB 与 PQ 的夹角为 90°。
PM = QM and PQ ⊥ AB
To check your construction, you can measure the distances from P to A and P to B (using the compass to compare lengths). They should be equal. Similarly, any point on the line PQ is equidistant from A and B.
要检验你的构造，你可以测量点 P 到 A 和 P 到 B 的距离（用圆规比较长度）。它们应当相等。同理，直线 PQ 上的任意一点到 A 和 B 的距离都相等。
4. Constructing an Angle Bisector | 构造角平分线
An angle bisector is a line or ray that divides an angle into two equal angles. The steps to construct the bisector of a given angle (say ∠ABC) are as follows:
角平分线是一条将角分成两个相等角的线或射线。构造给定角（例如∠ABC）的平分线的步骤如下：
Step 1: Place the compass point at the vertex B of the angle. Draw an arc that intersects both rays BA and BC. Label the intersection points as D and E respectively.
步骤一：将圆规尖端置于角的顶点 B。画一条弧线，使其与射线 BA 和 BC 都相交。将交点分别标记为 D 和 E。
Step 2: Without changing the compass width (or adjusting to a convenient width), place the compass point at D and draw an arc in the interior of the angle. Repeat with the compass point at E, using the same radius, to draw another arc that intersects the previous arc. Label the intersection point as F.
步骤二：保持圆规宽度（或调整到一个方便的宽度），将圆规尖端置于 D，在角的内部画一条弧线。然后将圆规尖端置于 E，使用相同半径画另一条弧线，与之前的弧线相交。将交点标记为 F。
Step 3: Draw the ray from B through F. This ray, BF, is the angle bisector of ∠ABC. Thus, ∠ABF = ∠FBC.
步骤三：从 B 出发画一条经过 F 的射线。这条射线 BF 就是 ∠ABC 的角平分线。因此，∠ABF = ∠FBC。
∠ABF = ∠FBC
Remember that the constructed ray splits the angle into two congruent angles. You can verify this by measuring or by folding along the bisector if you copy onto paper.
请记住，构造出的射线将角分成两个相等的角。你可以通过测量或沿平分线折叠来验证（如果画在纸上）。
5. Constructing a Perpendicular from a Point to a Line | 从一点到直线的垂线构造
Sometimes you need to drop a perpendicular from an external point to a given line. The construction ensures the shortest distance from the point to the line.
有时你需要从外部一点向给定直线作一条垂线。该构造
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