__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.

- 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
- Use the non-collapsing compass to construct the circle with radius AB and center C, intersecting circle O at D and E
- Construct line OD

## 4E solution[]

Given the triangle ABC.

- 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
- Construct line AC, intersecting OB at F
- Construct the circle with center O and radius OF, intersecting AC at G
- Construct line OG

## 3V solution[]

From the 3L solution:

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

From the 4E solution:

- Construct line BD, intersecting OC at H
- Construct the circle with center O and radius OH, intersecting BD at I
- Construct line OI
- Construct line CE, intersecting OD at J
- Construct the circle with center O and radius OJ, intersecting CE at K
- Construct line OK