Challenges From Ancient Greece

Holly Davis, Sep 7, 2026

[b]An Ancient Greek Challenge[/b] The study of Geometry was born in Ancient Greece, where mathematics was thought to be embedded in everything from music to art to governing the universe. Plato, an ancient philosopher and teacher had the statement, “Let no man ignorant of geometry enter here,” placed at the entrance of his school. This illustrates the importance of studying shapes and logic during that era. Everyone who learned geometry was challenged to construct geometric objects using two simple tools known as Euclidean tools: ● A straight edge without any markings (back of a ruler or paint stick) ● A compass The straight edge could be used to construct lines, and the compass to construct circles. As geometry grew in popularity, math students and mathematicians challenged each other to create constructions using only these two tools. Some constructions were reasonably easy (Can you construct a square?), some more challenging (Can you construct a regular pentagon?), and some impossible even for the greatest geometers (Can you trisect an angle? In other words, can you divide an angle into three equal angles?). Archimedes (287-212 B.C.E.) came close to solving the trisection problem, but his solution used a marked straight edge. Today, you will use GeoGebra compass and straightedge tools to explore geometric constructions as in the time of Euclid. [i]adapted from GaDOE Geometry CC Learning Plan: Challenges From Ancient Greece[/i]

Table of Contents

  1. Introduction
    1. Construction Introduction
  2. Segments
    1. Construct Congruent Segments
    2. Construct Perpendicular Bisector (Midpoint)
  3. Angles
    1. Construct Congruent Angles
    2. Construct Angle Bisectors
  4. Lines
    1. Construct Parallel Lines (3 ways)
    2. Construct Perpendicular Lines (not a bisector)
  5. Assessment
    1. Geometric Constructions Formative Assessment

1.

Introduction

Make sure you can use the available tools correctly.

  • 1. Construction Introduction

1.1.

Construction Introduction

Construction #1: Circle and Radius
Complete the following steps:[br]1) Using the POINT TOOL, place three points in the box below (Points A, B, and C)[br]2) Using the SEGMENT TOOL, draw segment AB[br]3) Using the COMPASS TOOL, create a circle with radius AB and center point C[br] [i]Tip: To make a circle with radius AB using the COMPASS TOOL, click on point A then point B[/i]
Using the MOVE TOOL, move point B in the problem above to change the radius of circle C.[br]How does the radius affect the circle?
Construction #2: Intersecting Circle and Line
Complete the following steps:[br]1) Using the COMPASS TOOL, create a circle with radius AB and centerpoint C[br]2) Using the POINT TOOL, mark points F and G at the intersections of circle C and segment DE[br]3) Using the SEGMENT TOOL, Draw triangle CFG
Construction #3: Intersecting Circles
Complete the following steps:[br]1) Using the COMPASS TOOL, create a circle with radius AB and centerpoint C[br]2) Using the COMPASS TOOL, create a circle with radius AB and centerpoint D [br]3) Using the SEGMENT TOOL, draw segment EF connecting the intersections of circle C and circle D
In the construction above, use the MOVE TOOL to move point B closer to point A. How does this affect line segment EF? What happens when the circles no longer intersect?
– Holly Davis, AJ Storck

2.

Segments

Construct a congruent segment and a segment bisector.

  • 1. Construct Congruent Segments
  • 2. Construct Perpendicular Bisector (Midpoint)

2.1.

Construct Congruent Segments

Follow these steps for congruent segment constructions:
1) Using the SEGMENT TOOL, make a segment CD of any length[br]2) Using the COMPASS TOOL, create a circle with radius AB and center point C[br]3) Using the POINT TOOL, mark the intersection of circle C and segment CD[br][br][i]REMEMBER: Congruent circles have the same radius. [i][i]The compass tool measures and preserves the radius. [/i][/i]Therefore, the [i]compass tool can be used to draw congruent segments.[/i][/i]
Construction #1
Construction #2
Construction #3
CHALLENGE QUESTION
Construct a segment that is TWICE the length as segment AB
– Holly Davis, AJ Storck

2.2.

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3.

Angles

Construct a congruent angle and an angle bisector.

  • 1. Construct Congruent Angles
  • 2. Construct Angle Bisectors

3.1.

Construct Congruent Angles

Directions
Use the tools provided to construct an angle congruent to the given angle[br][br]CONSTRUCTION STEPS:[br]1) Use the POINT TOOL to mark point F anywhere on segment AB[br]2) Use the COMPASS TOOL to create a circle with radius AF and center point A[br]3) Mark point G where circle A intersects segment AC[br]4) Use the COMPASS TOOL to create a circle with radius AF and center point D[br]5) Mark point H where circle D intersects the ray[br]6) Use the COMPASS TOOL to create a circle with radius FG and center point H[br]7) Mark point I where circle D and circle H intersect[br]8) Use the SEGMENT TOOL to draw segment DI
Construction #1
Construction #2
Construction #3
– Holly Davis, AJ Storck

3.2.

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4.

Lines

Construct parallel lines and perpendicular lines.

  • 1. Construct Parallel Lines (3 ways)
  • 2. Construct Perpendicular Lines (not a bisector)

4.1.

Construct Parallel Lines (3 ways)

Follow these steps to construct parallel lines by creating corresponding angles
1) Using the POINT TOOL, mark point H anywhere on segment FB [i](Hint: FH must be shorter than FG)[/i][br]2) Using the COMPASS TOOL, create a circle with radius FH and center point F[br]3) Using the POINT TOOL, mark point I at the intersection of circle F and segment FG[br]4) Using the COMPASS TOOL, create a circle with radius FH and center point G[br]5) Using the POINT TOOL, mark point J at the intersection of circle G and segment GC[br]6) Using the COMPASS TOOL, create a circle with radius HI and center point J[br]7) Using the SEGMENT TOOL, draw a segment from G to the intersection of circles G and J
Construction #1: Corresponding Angle Method
Follow these steps to construct parallel lines by creating congruent triangles
1) Using the COMPASS TOOL, create a circle with radius DG and center point F.[br]2) Using the POINT TOOL, mark point H at the rightmost intersection of circle F and segment AB.[br]3) Using the COMPASS TOOL, create a circle with radius CD and center point F.[br]4) Using the COMPASS TOOL, create a circle with radius CG and center point H.[br]5) Using the SEGMENT TOOL, draw a segment from C to the intersection of circles F and H.
Construction #2: Triangle Method
Follow these steps to construct parallel lines by creating a rhombus
1) Using the COMPASS TOOL, create a circle with radius CD and center point C.[br]2) Using the POINT TOOL, mark point F at the bottom right intersection of circle C and segment AB (the closest intersection to point B).[br]3) Using the COMPASS TOOL, create a circle with radius CF and center point F.[br]4) Using the POINT TOOL, mark point G at the intersection of circle F and segment AB.[br]5) Using the COMPASS TOOL, create a circle with radius CF and center point G[br]6) Using the POINT TOOL, mark point H at the intersection of circle G and circle C.[br]7) Using the SEGMENT TOOL, draw a segment from C to H.
Construction #3: Rhombus Method
Which method do you prefer?
Why do you prefer this method?
– Holly Davis, AJ Storck

4.2.

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5.

Assessment

Test your new knowledge about constructions.

  • 1. Geometric Constructions Formative Assessment

5.1.

Geometric Constructions Formative Assessment

1) Which of the following tools can be used to complete a formal geometric construction? Select all that apply
2) On a compass, what does the opening between the center point to the pencil point represent?
3) Fill in the blank: Given two congruent circles, the distance from the centerpoint to the edge of each circle is __________ congruent.
4) Explain your reasoning to Question 3
5) Why is it important that the term "locked" is used when making the following statement?[br][br] You can use a [b]locked[/b] compass setting and a straightedge to draw two congruent segments.
– Holly Davis, AJ Storck

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