The following is a collection of problems from Hartshorne’s Geometry: Euclid and Beyond that involve ruler and compass constructions. The problems are based on Books I, III, and IV of Euclid’s Elements. Each problem has a specified number of steps (denoted as par) that is the minimum number of steps required to complete the construction. Each of the proofs uses the optimal par.

Counting Steps #

What is counted as a step: #

• Using the ruler to draw a new line through two distinct points (given or previously constructed)
• Using the ruler to extend a given or previously constructed line in either direction
• Using the compass to draw a new circle with center at a given or constructed point and radius equal to the distance between two given or constructed points

What is not counted as a step: #

• Extending lines that are already given or constructed
• Choosing points at random or subject to conditions, such as lying on a given line or circle
• Obtaining new points as intersections of lines and circles - these points are considered constructed automatically

The ruler may not be used to measure distances or have any markings (it is a straightedge only).

2.1 Given an angle, construct the angle bisector #

par = 4

2.2 Given a line segment, find the midpoint of that segment #

par = 3

2.3 Given a line $l$ and a point $A$ on $l$, construct a line perpendicular to $l$ through $A$ #

par = 4, possible in 3

2.4 Given a line $l$ and a point $A$ not on $l$, construct a line perpendicular to $l$ passing through $A$ #

par = 4, possible in 3

2.5 Given an angle at a point $A$, and given a ray emanating from a point $B$, construct an angle at $B$ equal to the angle at $A$ #

par = 4

2.6 Given a line $l$ and a point $A$ not on $l$, construct a line parallel to $l$ passing through $A$ #

par = 3

2.7 Given the circumference of a circle, find the center of the circle #

par = 5

2.8 Given a circle with center $O$, and a point $A$ outside the circle, construct a line through $A$ tangent to the circle. #

par = 6

Warning: You may not slide the ruler until it seems to be tangent to the circle. You must construct another point on the desired tangent line before drawing the tangent.

2.9 Construct a circle inscribed in a given triangle $ABC$ #

par = 13

2.10 Construct a circle circumscribed about a given triangle $ABC$ #

par = 7

2.11 Given a line $l$, a line segment $d$, and a point $O$, construct a circle with center $O$ that cuts off a segment congruent to $d$ on the line $l$ #

par = 9

2.12 Given a point $A$, a line $l$, and a point $B$ on $l$, construct a circle that passes through $A$ and is tangent to the line at $B$ #

par = 8

2.13 Construct three circles, each one meeting the other two at right angles. #

par = 11

We say that two circles meet at right angles if the radii of the two circles to a point of intersection make right angles.