Foundations of Geometry - Constructions

Gianluca Occhetta, 6 Apr 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

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.

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1.3.

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1.4.

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1.5.

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1.6.

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1.7.

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1.8.

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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.

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2.3.

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2.4.

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2.5.

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2.6.

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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.

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3.3.

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3.4.

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3.5.

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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.

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4.3.

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4.4.

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4.5.

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