Instruction: Construct a rhombus with the given side and an angle of 45° in a vertex.

Goal: 5L 7E

Available tools:

• Move
• Point
• Line
• Circle
• Perpendicular Bisector
• Perpendicular
• Angle Bisector
• Intersect

Pack: Gamma

Previous level: 3.7

Next level: 3.9

## 5L solution

Given the side AB.

1. Construct the perpendicular to AB from A; let C be a distinct arbitrary point on that perpendicular
2. Construct the angle bisector of ∠BAC
3. Construct the circle with center A and radius AB, intersecting the angle bisector at D and line AC at E with E on the opposite side of C from A
4. Construct the perpendicular to AC through D
5. Construct line BE

## 7E solution

Given the side AB.

1. Construct the circle with center A and radius AB
2. Construct the circle with center B and radius AB, intersecting circle A at C and D
3. Construct line CD, intersecting AB at E
4. Construct the circle with center E and radius AE, intersecting CD at F and G
5. Construct line AF, intersecting circle A at H
6. Construct line BG, intersecting circle B at I on the same side as H relative to AB
7. Construct line HI

## 4V solution

From the 5L solution:

1. Construct the perpendicular to AD through B
2. Construct the perpendicular to AD from A; let F be the diametrical opposite of the intersection of circle A with the perpendicular to AC through D on circle A
3. Construct the perpendicular to CD through F

From the 7E solution: Let J be the other intersection of line AF with circle A and K the other intersection of line BG with circle B.

1. Construct line JK
2. Construct line AG
3. Construct line BF