# Problems

Here are a few more construction problems to try your hand at.

1. Given a semicircle centered at a point with diameter , find points and on , and points and on the semicircle such that the quadrilateral is a square.

2. Given a quadrant of a circle (two radii that make an angle of and the included arc), construct a new circle that is inscribed in the quadrant (in other words, the new circle is tangent to both rays and to the quarter arc of the quadrant).

3. Given a point , a line that does not pass through , and a point on , construct a circle passing through that is tangent to at the point .

4. Given two points and that both lie on the same side of a line , find a point on such that and make the same angle with .

5. Given two points and that both lie on the same side of line , find a point on such that is as small as possible. (Hint: This problem is related to the construction problem 4. Also remember that the shortest distance between two points is a line.)

6. Given two non-parallel lines and and a radius , construct a circle of radius that is tangent to both and .