Basic Constructions

Here are a list of elementary constructions that can be carried out with a straightedge and compass. They are arranged roughly in order of difficulty, and if you can do most of these, you can do most standard geometric constructions.

1. Copy a segment. In other words, mark off a segment that exactly matches the length of a given segment on a different straight line.

2. Copy an angle. Given an angle, make another angle of exactly the same size somewhere else.

3. Bisect a segment. This problem was already solved in Section 1

4. Bisect an angle. Given an angle, find a line through the vertex that divides it in half.

5. Construct a line perpendicular to a given line through a point on the given line.

6. Construct a line perpendicular to a given line and passing through a point not on the given line.

7. Given a line $L$ and a point $P$ not on $L$, construct a new line that passes through $P$ and is parallel to $L$.

8. Construct an angle whose size is the sum or difference of two given angles.

9. Given three segments, construct a triangle whose sides have the same lengths as the segments.

10. Construct the perpendicular bisector of a line segment.

11. Given three points, construct the circle that passes through all of them.

12. Given a circle, find its center.

13. Given a triangle $T$, construct the inscribed and circumscribed circles. The inscribed circle is a circle that fits inside the triangle and touches all three edges; the circumscribed circle is outside the triangle except that it touches all three of the vertices of the triangle.

14. Construct angles of $90^{\circ}$, $45^{\circ}$, $30^{\circ}$, $60^{\circ}$, and if you want a challenge, $72^{\circ}$.

15. Construct a regular pentagon. (A regular pentagon is a five sided figure all of whose sides and angles are equal.)

16. Given a point $P$ on a circle $C$, construct a line through $P$ and tangent to $C$.

17. Given a circle $C$ and a point $P$ not on $C$, construct a line through $P$ and tangent to $C$.

18. Given two circles $C_1$ and $C_2$, find lines internally and externally tangent to both.