Study Notes: Constructions

Overview

Constructions is a hands-on geometry topic that tests your ability to use only a compass and straight-edge (unmarked ruler) to draw precise geometric figures. In SOF IMO, you won't physically draw constructions during the exam, but you must understand the **logical steps**, the **order of operations**, and the **geometric principles** behind each construction. Questions typically ask you to identify the correct sequence of steps, recognize which construction method applies, or determine properties of constructed figures.

This topic bridges pure geometry with practical problem-solving. Mastering constructions strengthens your understanding of angle bisectors, perpendicular bisectors, triangle properties, and tangent-circle relationships—concepts that appear across multiple IMO sections. Focus on memorizing the step sequences and understanding *why* each step works, not just mechanical reproduction.

The most common constructions tested are: bisecting angles and line segments, constructing perpendiculars, drawing triangles from given data (SSS, SAS, ASA, RHS criteria), and drawing tangents to circles from external points. Expect 2–4 questions in the exam, often in the Achievers Section as multi-step reasoning problems.

Key Concepts

• **Compass and straight-edge only**: All constructions must be done without measuring angles or lengths. The compass transfers distances; the straight-edge connects points. No protractor or marked ruler allowed.

• **Arc intersections create key points**: Most constructions rely on drawing arcs from two different centers. Where arcs intersect defines new points with specific geometric properties (equidistant from centers, for example).

• **Angle bisector divides an angle into two equal parts**: Constructed by drawing equal arcs from the angle vertex, then arcs from where those arcs cut the arms, creating an intersection that lies on the bisector.

• **Perpendicular bisector of a segment**: Every point on this line is equidistant from the segment's endpoints. Constructed by arcs of equal radius from both endpoints; their intersections determine the perpendicular bisector.

• **Triangle construction uses congruence criteria**: SSS (three sides), SAS (two sides and included angle), ASA (two angles and included side), RHS (right angle, hypotenuse, one side). Each criterion has a specific construction sequence.

• **Tangent from external point**: A tangent is perpendicular to the radius at the point of contact. Constructing tangents requires finding the perpendicular from the external point to the radius, often using a semicircle on the line joining the point to the center.