Problems

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

1. Given a semicircle centered at a point $C$ with diameter $AB$, find points $I$ and $J$ on $AB$, and points $H$ and $G$ on the semicircle such that the quadrilateral $GHIJ$ is a square.

2. Given a quadrant of a circle (two radii that make an angle of $90^\circ$ 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 $A$, a line $L$ that does not pass through $A$, and a point $B$ on $L$, construct a circle passing through $A$ that is tangent to $L$ at the point $B$.

4. Given two points $A$ and $B$ that both lie on the same side of a line $L$, find a point $C$ on $L$ such that $AC$ and $BC$ make the same angle with $L$.

5. Given two points $A$ and $B$ that both lie on the same side of line $L$, find a point $C$ on $L$ such that $AC + BC$ 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 $L_1$ and $L_2$ and a radius $r$, construct a circle of radius $r$ that is tangent to both $L_1$ and $L_2$.