Plane Geometry - Bruce E. Shapiro

Section 45Euclidean ConstructionsThis section outlines the basic Euclidean constructions that use a straightedgeand a compass (or the straight-edge and circle tools in GeoGebra).Construction 45.1 (Copy a Line Segment) Given a line segment AB,and a ray −−→ CD, find a point E ∈ −−→ CD such that AB = CE.1. Place the ends of a the compass at points A and B to determine itslength AB. [ In GeoGebra, use the line segment tool to create asegment AB and observe its name (say it is called b).]2. With the compass fixed, place one end at C and draw a circular arcintersecting −−→ CD of radius AB. [ In GeoGebra use the “Circle withcenter and given radius,” then click on point C. When the pop-upmenu asks you for the value of the radius, type the letter b, which isthe name of segment AB.]3. Define E as the intersection of the new circle with −−→ CD. [ In GeoGebra,using the “Intersection of two objects” tool, click first anywhere onthe circle, and then anywhere on ray −−→ CD. Point E will appear.]4. Construct segment CE. [In Geogebra, use the “Segment between twopoints” and click first on C and then on E.]Construction 45.2 (Copy an Angle) Given an angle α = ∠BAC and aray −−→ DE, construct an angle ∠EDF that is congruent to α.1. Construct a circle d = C(A, AB).2. Use the compass to measure the distance AB.3. Construct a circle h = C(D, AB). This circle intersects −→ AE at Q.251