Challenges from Ancient Greece
The study of Geometry was born in Ancient Greece, where principles of mathematics were understood to provide the basis of musical harmony, determine the beauty of art, and govern the nature of the universe. Mathematics was central to Greek philosophy and religion. Plato, an ancient philosopher and teacher, had the following statement placed at the entrance of his school: “Let no man ignorant of geometry enter here”. The statement illustrates the importance of the study of shapes and logic during that era.
Students of geometry were challenged to construct geometric objects using only two simple tools.
Known as Euclidean tools, they were a compass and a straight edge. The compass was used to establish distance, and the straightedge to establish collinearity. All geometric constructions are based on these two concepts.
The compass does not simply draw curves. It cannot be replaced by a circular template, such as a coffee can or jar. The compass is anchored at a center point, and keeps its pencil at a fixed distance from that point. All points on the curve drawn by a compass are equidistant from the center point.
Although rulers are often used as straightedges, a true Euclidean straightedge has no graduation marks. The straightedge is used only for drawing lines. This instrument can be used to draw the set of all points that are collinear with any two distinct points.
Rulers and protractors have their place in geometry, but they are not construction instruments. All measurements are approximations. In reality, the compass and straightedge also are subject to error.
A line has zero width, but a drawing of a line does not. However, construction is a theoretical exercise. If we were not limited by the physical imperfections of the instruments, the drawing surface, and the person using them, then construction instruments would render exact results. This is not true of measurement instruments.
Some constructions are fairly easy (constructing a square), some more challenging (constructing a pentagon), and some are impossible (trisecting an angle), even for the greatest geometers.
Archimedes (287-212 B.C.E) came close to solving the trisection problem, but his solution used a marked straight edge.

Geometric Constructions: Use the notes provided to complete the following constructions.
1.
Copy a line segment
2.
Copy an angle
3.
Bisect a line segment

4.
Bisect an angle
5.
Construct a line segment perpendicular to a given line through a given point P on the line segment.

6.
Construct the perpendicular bisector of a line segment.
7.

8.
Construct a line segment parallel to a given segment through a point not on the line.
9.
Construct an equilateral triangle inscribed in a circle.

10.
Construct a square inscribed in a circle.
11.
Construct a regular hexagon inscribed in a circle.

12.
Construct the inscribed circle of the triangle.
13.
Construct the circumscribed circle of the triangle.