6.10 Symmetry of Four Lines

Instruction: Three lines are intersected in a point. Construct a line so that the set of all 4 lines is mirror symmetric.

Goal: 3L 4E

Available tools:

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

Pack: Zeta

Previous level: 6.9

Next level: 6.11

3L solution

Let O be the intersection of the three lines.

1. Construct a circle with center O and an arbitrary radius, intersecting the lines at six points; let A, B and C be a set of three consecutive points
2. Use the non-collapsing compass to construct the circle with radius AB and center C, intersecting circle O at D and E
3. Construct line OD

4E solution

Given the triangle ABC.

1. Construct a circle with center O and an arbitrary radius, intersecting the lines at six points; let A, B, C, D amd E be a set of five consecutive points
2. Construct line AC, intersecting OB at F
3. Construct the circle with center O and radius OF, intersecting AC at G
4. Construct line OG

3V solution

From the 3L solution:

1. Construct line OE
2. Use the non-collapsing compass to construct the circle with radius BC and center A, intersecting circle O at F
3. Construct line OF

From the 4E solution:

1. Construct line BD, intersecting OC at H
2. Construct the circle with center O and radius OH, intersecting BD at I
3. Construct line OI
4. Construct line CE, intersecting OD at J
5. Construct the circle with center O and radius OJ, intersecting CE at K
6. Construct line OK