7.8 Circle Tangent to Three Lines

Instruction: Construct a circle that is tangent to the three given lines. Two of the lines are parallel.

Goal: 4L 6E

Available tools:

• Move
• Point
• Line
• Circle
• Perpendicular Bisector
• Perpendicular
• Angle Bisector
• Parallel
• Non-collapsing Compass
• Intersect

Pack: Eta

Previous level: 7.7

Next level: 7.9

4L solution

Given the intersections of the lines A and B.

1. Construct the angle bisector of ∠CAB, with C an arbitrary point on one of the parallel lines.
2. Construct the angle bisector of ∠ABD, with D an arbitrary point on the other parallel line and on the same side as C from the third given line, intersecting the angle bisector of ∠CAB at E
3. Construct the perpendicular to AB through E, intersecting AB at F
4. Construct the circle with center E and radius EF

6E solution

Given the intersections of the lines A and B.

1. Construct the circle with center A and radius AB, intersecting the parallel line through A at C and D
2. Construct line BC
3. Construct the circle with center C and radius BC
4. Construct the circle with center D and radius CD, intersecting circle C at E and F
5. Construct line EF, intersecting BC at G and AC and H
6. Construct the circle with center G and radius GH

2V solution

From the 4L solution:

1. Construct the angle bisector of ∠GAB, with G an arbitrary point on AC and on the other side of C from A.
2. Construct the angle bisector of ∠ABH, with H an arbitrary point on BD and on the other side of D from B, intersecting the angle bisector of ∠GAB at I
3. Construct the perpendicular to AB through I, intersecting AB at J
4. Construct the circle with center I and radius IJ