## Problem 6, (3/6 - 3/24) - Compass and straight-edge

In compass-and-straight-edge construction problems, we are allowed to make the following legal moves:

1) We can connect two points with a straight line.

2) We can find the point(s) of intersection of any two straight lines, a line and a circle, or two circles.

3) We can transfer lengths with the compass.

4) We can extend an existing line to arbitrary length.

5) We can construct a circle with a given center and radius.

Note that by a straight-edge, we do *not* mean a ruler - that is, a straight-edge cannot be used to measure arbitrary lengths. It is merely a straight line, with no quantitative or metric properties associated to it.

Using the aforementioned rules, a compass (if you don't have one handy, a pencil tied to a piece of string works just fine) and a straight-edge (a ruler, the edge of a book...):

a) (Easy) Duplicate an arbitrary angle.

b) (Easy) Bisect an arbitrary angle.

c) (Easy) Construct the perpendicular to a given line through a given point on the line.

d) (Medium) Construct the perpendicular to a given line through a given point off the line.

e) (Medium) Construct the parallel to a given line through a given point off the line.

f) (Bonus points) Given a circle, its center, and a point off the circle, construct the line tangent to the circle through the given point.

Any submitted solution should be verifiable through various properties of Euclidean geometry - no "eyeballing" allowed. Part f) is for bonus points only - any solution that correctly solves a) through e) will be counted as a correct solution.

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