Foundations of Geometry - Constructions

Gianluca Occhetta, Apr 6, 2020

Table of Contents

  1. Sheet 1
    1. Exercise 2.1 - Angle bisector
    2. Exercise 2.2 - Midpoint of a segment
    3. Exercise 2.3 - Perpendicular from an inner point
    4. Exercise 2.4 - Perpendicular from an outer point
    5. Exercise 2.5 - Trasporting an angle
    6. Exercise 2.6 - Parallel by a given point
    7. Exercise 2.7 - Find the center of a given circle
    8. Exercise 2.8 - Tangent to a circle
  2. Sheet 2
    1. Exercise 2.9 - Finding the incenter
    2. Exercise 2.10 - Finding the circumcenter
    3. Exercise 2.11 - Circle which cuts a given chord on a given line
    4. Exercise 2.12 - Circle by a point and tangent to a line at a given point
    5. Exercise 2.13 - Construct three orthogonal circles
    6. Exercise 2.14 - (Quick) segment trisection
  3. Sheet 3
    1. Exercise 10.1
    2. Exercise 10.2
    3. Exercise 10.3
    4. Exercise 10.4
    5. Exercise 10.5
  4. Sheet 4
    1. Exercise 2.16 - The rusty compass
    2. Exercise 2.17 - Perpendicular from a point on the line (with rusty compass)
    3. Exercise 2.18 - Perpendicular from a point not on the line (with rusty compass)
    4. Exercise 2.19 - Transporting segments with rusty compass
    5. Exercise 2.20 - Circle/line intersection with rusty compass - II
  5. Sheet 5
    1. Exercise - 2.24 a) - External common tangents
    2. Exercise 2.24 b) -Internal common tangents
    3. Exercise 3.5 - Circles tangent to a given line and passing by two points
    4. Exercise 3.6 - Circle tangent to two given lines and passing by a point
    5. Elements III.33
  6. Sheet 6
    1. Exercise 4.3 - Inscribed equilateral triangle
    2. Exercise 4.5 - Regular pentagon, given a side
    3. LLC 1 - Parallel lines
    4. LLC 2 - Non parallel lines

1.

Sheet 1

  • 1. Exercise 2.1 - Angle bisector
  • 2. Exercise 2.2 - Midpoint of a segment
  • 3. Exercise 2.3 - Perpendicular from an inner point
  • 4. Exercise 2.4 - Perpendicular from an outer point
  • 5. Exercise 2.5 - Trasporting an angle
  • 6. Exercise 2.6 - Parallel by a given point
  • 7. Exercise 2.7 - Find the center of a given circle
  • 8. Exercise 2.8 - Tangent to a circle

1.1.

Exercise 2.1 - Angle bisector

3 circles + 1 line = 4 steps (par 4)
– Gianluca Occhetta

1.2.

–

1.3.

–

1.4.

–

1.5.

–

1.6.

–

1.7.

–

1.8.

–

2.

Sheet 2

  • 1. Exercise 2.9 - Finding the incenter
  • 2. Exercise 2.10 - Finding the circumcenter
  • 3. Exercise 2.11 - Circle which cuts a given chord on a given line
  • 4. Exercise 2.12 - Circle by a point and tangent to a line at a given point
  • 5. Exercise 2.13 - Construct three orthogonal circles
  • 6. Exercise 2.14 - (Quick) segment trisection

2.1.

Exercise 2.9 - Finding the incenter

Construct a circle inscribed in a given triangle ABC (par = 13).[br][br]10 circles + 3 lines = 13 steps.
– Gianluca Occhetta

2.2.

–

2.3.

–

2.4.

–

2.5.

–

2.6.

–

3.

Sheet 3

  • 1. Exercise 10.1
  • 2. Exercise 10.2
  • 3. Exercise 10.3
  • 4. Exercise 10.4
  • 5. Exercise 10.5

3.1.

Exercise 10.1

– Gianluca Occhetta

3.2.

–

3.3.

–

3.4.

–

3.5.

–

4.

Sheet 4

  • 1. Exercise 2.16 - The rusty compass
  • 2. Exercise 2.17 - Perpendicular from a point on the line (with rusty compass)
  • 3. Exercise 2.18 - Perpendicular from a point not on the line (with rusty compass)
  • 4. Exercise 2.19 - Transporting segments with rusty compass
  • 5. Exercise 2.20 - Circle/line intersection with rusty compass - II

4.1.

Exercise 2.16 - The rusty compass

4 circles + 2 lines = 6 steps (par 6)
The triangles ACD and BEF are congruent by (I.8). It follows that ABH is similar to ACD by SIM AAA, hence ABH is equilateral
– Gianluca Occhetta

4.2.

–

4.3.

–

4.4.

–

4.5.

–

5.

Sheet 5

  • 1. Exercise - 2.24 a) - External common tangents
  • 2. Exercise 2.24 b) -Internal common tangents
  • 3. Exercise 3.5 - Circles tangent to a given line and passing by two points
  • 4. Exercise 3.6 - Circle tangent to two given lines and passing by a point
  • 5. Elements III.33

5.1.

Exercise - 2.24 a) - External common tangents

Ex. 2.24 a)
– Gianluca Occhetta

5.2.

–

5.3.

–

5.4.

–

5.5.

–

6.

Sheet 6

  • 1. Exercise 4.3 - Inscribed equilateral triangle
  • 2. Exercise 4.5 - Regular pentagon, given a side
  • 3. LLC 1 - Parallel lines
  • 4. LLC 2 - Non parallel lines

6.1.

Exercise 4.3 - Inscribed equilateral triangle

The triangles ABD and ABC are equilateral, hence the angle ABE is 120°.[br]The angle ADF is 120°, too, since ADB is 60°.[br]Moreover the angles AFB and AEF are congruent, since they insist on the same arc. Finally AD is congruent to AB, so the triangles ADF and AEB are congruent, so AE is congruent to AF.[br][br]By construction the angle FBE is 60°, so also EAF is 60°, since they insist upon the same arc. So also the angles AEF and AFE are 60°
– Gianluca Occhetta

6.2.

–

6.3.

–

6.4.

–

Information