Essential Geometric Constructions

Why This Matters

Geometric constructions aren't just exercises in compass-and-straightedge manipulation—they're the foundation for understanding why geometric relationships work, not just that they work. In Discrete Geometry, you're being tested on your ability to recognize underlying principles: equidistance, angle relationships, symmetry, and locus of points. Every construction you learn demonstrates a theorem in action, and exam questions will ask you to connect the procedure to the property it guarantees.

Think of constructions as proofs you can draw. When you bisect an angle, you're proving that a ray exists which creates two congruent angles. When you find a circumcenter, you're proving that three perpendicular bisectors are concurrent. Don't just memorize the steps—know what geometric principle each construction illustrates and why that principle matters for solving problems.

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Bisector Constructions: Finding Equidistant Points

Bisector constructions rely on a single powerful idea: every point on a bisector is equidistant from the objects being bisected. This equidistance property is what makes bisectors so useful for locating special points in triangles and other figures.

Perpendicular Bisector of a Line Segment

• Creates the locus of all points equidistant from two endpoints—this is the fundamental property that makes circumcenters possible
• Compass arcs from both endpoints must have equal radii; their intersections define two points on the bisector
• Guarantees both perpendicularity and equal division—the bisector hits the segment at a $90°$ angle and splits it into congruent parts

Angle Bisector

• Creates the locus of all points equidistant from two rays—measured as perpendicular distance to each side
• Arc from the vertex intersects both rays; equal arcs from those intersections locate a point on the bisector
• Divides any angle into two congruent angles—essential for constructing incenters and solving angle-chase problems

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Line Constructions: Parallel and Perpendicular Relationships

These constructions establish the fundamental relationships between lines. Parallel lines maintain constant separation; perpendicular lines intersect at exactly $90°$. Both constructions depend on angle replication or the properties of circles.

Parallel Line Through a Point

• Uses corresponding angles or alternate interior angles—replicating an angle guarantees parallelism by the converse of the parallel postulate
• Requires a transversal through the external point to create a reference angle with the original line
• Proves the Euclidean parallel postulate in action—exactly one parallel exists through any external point

Perpendicular Line Through a Point

• Two cases exist: point on the line vs. point off the line—the construction differs slightly but uses the same principle
• Arc intersections create isosceles triangles—the perpendicular passes through the apex, exploiting symmetry
• Forms a $90°$ angle verified by the fact that the constructed point is equidistant from two points on the original line

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Regular Polygon Constructions: Exploiting Symmetry

Regular polygons have equal sides and equal angles, which means their vertices lie on a circle and divide it into equal arcs. Constructions leverage this circular symmetry—some polygons (like hexagons) divide naturally with a compass, while others (like squares) require perpendicular constructions.

Equilateral Triangle

• All sides equal to the compass radius—arcs from each endpoint with radius equal to the segment length intersect at the third vertex
• Each interior angle measures $60°$—a consequence of three equal sides forcing three equal angles
• Simplest regular polygon to construct—requires only the original segment length, no angle measurement

Square

• Requires perpendicular construction at each endpoint—you cannot construct a $90°$ angle with compass alone without using perpendicularity
• All four sides equal, all four angles $90°$—mark the segment length along each perpendicular to locate vertices
• Demonstrates that squares are special rectangles—the construction builds right angles first, then enforces equal sides

Regular Hexagon

• Radius equals side length—this unique property makes hexagons constructible by simply "walking" the compass around the circle
• Circle divided into six $60°$ arcs—each central angle spans one-sixth of $360°$
• Contains six equilateral triangles—connecting all vertices to the center reveals the hexagon's internal structure

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Triangle Centers: Concurrence of Special Lines

Every triangle has special points where certain lines meet. The remarkable fact is that these lines always concur—three perpendicular bisectors meet at one point, three angle bisectors meet at another. Each center has a defining equidistance property.

Circumcenter

• Intersection of perpendicular bisectors—only two are needed since all three are guaranteed to be concurrent
• Equidistant from all three vertices—this distance is the circumradius $R$
• Center of the circumscribed circle—the unique circle passing through all three vertices; can lie outside the triangle for obtuse triangles

Incenter

• Intersection of angle bisectors—again, two suffice due to guaranteed concurrence
• Equidistant from all three sides—this distance is the inradius $r$, measured perpendicularly to each side
• Center of the inscribed circle—the unique circle tangent to all three sides; always lies inside the triangle

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Tangent Constructions: Circle-Line Relationships

A tangent line touches a circle at exactly one point, and the key property is that the radius to the point of tangency is perpendicular to the tangent. This $90°$ relationship is what makes the construction possible.

Tangent Line from an External Point

• Connect external point to center, then find the midpoint of this segment—the midpoint becomes the center of a new construction circle
• Thales' theorem guarantees the right angle—any point on a circle with diameter $\overline{PC}$ (where $P$ is external, $C$ is center) forms a $90°$ angle
• Two tangent lines exist from any external point—they have equal length from the external point to the points of tangency

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Quick Reference Table

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Self-Check Questions

1. Which two constructions both rely on the principle that points on a bisector are equidistant from two objects? What distinguishes what they're equidistant from?

2. If you need to find a point equidistant from all three sides of a triangle, which construction do you use? What if you need equidistance from all three vertices?

3. Compare and contrast the construction of an equilateral triangle with the construction of a regular hexagon. Why does the same compass setting work for both?

4. A problem asks you to construct a circle passing through three non-collinear points. Which triangle center do you need, and what construction produces it?

5. Explain why the tangent line construction requires finding a midpoint and drawing an auxiliary circle. What theorem guarantees that this process creates a $90°$ angle at the point of tangency?