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Just like the ancient Greeks, a good geometry requires building pristine objects with the simplest tools. Can you create these key characters?

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Midpoints & Perpendicular Bisectors

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The content on this page is supplementary material for Marco the Great and the Shape of Everything.

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Geometer's Toolbox

If we want our Geometry to be perfect, like Euclid, we cannot rely on imperfect tools such as rulers. Measurement is always imprecise. Give it a try! Make a segment that is exactly 1 inch long. Can you do it? While you may have pulled out a ruler and marked the key points, your segment is not exactly 1 inch.

Despite human advancements in technology and manufacturing, making a "perfect ruler" remains a challenge. Most companies even state their measuring devices are accurate to 1/16 of an inch and that doesn't account for human error. When you draw your 1 inch segment, you aren't likely to stop at exactly 1 inch. Maybe 1.1 or 1.01, but hitting the target perfectly is nearly impossible.

Thus, in building a geometry we rely only on two tools - Tools that ensure perfection or at least get us as close as we can get.

The Tools

Collapsible Compass

Used to make circles, a collapsible compass does not hold length. When you pick it up, it folds back down. You can make your own at home by tying a string to a pencil.

Straightedge

Used to connect points or extend segments with crisp marks, a straightedge is like a ruler, but does not contain any markings. You can't use a straightedge to measure.

Pencil

We can use our pencil in tandem with our other tools to erect circles, segments, and representations of lines and rays, as well as mark off points and the intersection of our objects.

Valid Moves

What can we do with these tools?

Points & Intersections

Mark off a point at the intersection of two or more objects by selecting two objects, or marking the location of the intersection.

Segments, Rays, & Lines

Connect two points with a segment, or extend a segment to form a line or a ray by selecting two points.

Circles

Create a circle by selecting a center point and a point on the circle.

Equilateral Triangle

Euclid's very first step in building his geometry was to construct an Equilateral Triangle. While it may seem like an odd choice, perhaps Euclid had a bit of foresight. The Equilateral Triangle is a mighty beast! This simple construction would pave the way for many key characters in our story.

An Equilateral Triangle is a triangle where each of the three sides have equal length.

Build It

Use the Geometer's Toolbox to construct a perfect Equilateral Triangle. You are given one side to start with. The activity will indicate if you are successful.

©MathBait® created with GeoGebra

The Solution

Construct a circle centered at A passing through B
Construct a circle centered at B passing through A
Mark the intersection point of the two circles as C
Create polygon ΔABC

The Proof

Since C is on the circle centered at A, AC is a radius. And, as B is on the circle centered at A, AB is a radius. This tells us AC=AB (by definition of a circle).

Similarly, as both A and C are on the circle centered at B, AB and BC are radii. Thus, AB=BC.

We have AC=AB=BC making ΔABC an equilateral triangle.

Midpoints and Perpendicular Bisectors

With an Equilateral Triangle, a whole new world of possibilities emerges. This special triangle holds a number of secrets we can use to our advantage.

The midpoint of a segment AB is the point, M, on AB such that AM≅MB.

Build It

Use the Geometer's Toolbox to construct the midpoint of the given segment. The activity will indicate if you are successful.

©MathBait® created with GeoGebra

The Solution

Construct a circle centered at A passing through B
Construct a circle centered at B passing through A
Mark the intersection points of the two circles as C and D
Connect points C and D with your straightedge
Mark the intersection point of AB and CD
You found the midpoint!

Why it Works

While this point may look like the midpoint, proving it is exactly halfway between A and B is more challenging. We can't rely on what we think to be true or even what appears to be true (maybe M is 1.2 cm from A and 0.8 cm from B, too small a difference for our eyes to notice).

Since an Equilateral triangle has symmetry, there is a line that cuts our shape perfectly in half. Our midpoint must lie on this line as it is the halfway point of the base. You can find the full proof using our axioms and previously proven theorems in The Shape of Everything.

The symmetry of our Equilateral triangle will come in handy again to construct perpendicular bisectors. While most of our geometric objects have only one superpower, this Avenger has two! It not only finds the point that cuts a segment in half, it also cuts it to make perfect right angles.

The perpendicular bisector of a segment AB is the line passing through the midpoint of AB and perpendicular to AB.

Build It

Use the Geometer's Toolbox to construct the perpendicular bisector of the given segment. The activity will indicate if you are successful. Hint the perpendicular bisector is a line. Use the line tool to complete your build.

©MathBait® created with GeoGebra

The Solution

Construct a circle centered at A passing through B
Construct a circle centered at B passing through A
Mark the intersection points of the two circles as C and D
Connect points C and D with your straightedge to create a line
You found the perpendicular bisector!

Perpendicular Lines

Two lines are perpendicular if they meet at a right angle (to form an L). While the perpendicular bisector is powerful, can we extend this idea to create a perpendicular to a given line at any point?

Two lines are perpendicular if they meet at a right angle.

Build It

Use the Geometer's Toolbox to construct a line perpendicular to the given segment that passes through the given point. The activity will indicate if you are successful.

©MathBait® created with GeoGebra

The Solution

Extend the line passing through points A and B
Construct a circle centered at C intersecting line AB at two points
Mark the intersection of your circle and line AB, call these points D and E
Using your previous constructions, construct the perpendicular bisector of DE
This line is perpendicular to AB (since lines AB and DE are the same line!) and passes through point C!

Why it Works

It seems strange we can draw any circle centered at C to find this perpendicular line! So what is happening here? When we draw a circle centered at C, we know every point on this circle is the same distance from C (definition of circle). This means our two intersection points, D and E, are the same length: CD=CE.

Our resulting triangle, ΔCDE, is a special character - an Isosceles triangle. An Isosceles triangle also has symmetry (though not as powerful as an Equilateral triangle.) This allows us to construct the perpendicular bisector of DE which passes through C, cutting our Isosceles triangle in half.

Angle Bisectors

You might have noticed we are cutting a lot of our characters in half. Why? When it comes to angles, halving allows us to make new and useful characters in our story of geometry. If we can construct a right angle, 90°, halving it means we can now also create a 45° angle! We can also use our Equilateral triangles to build 60° angles, and halving will give us 30°, and 15° as well. Much of the power of geometry lies in using what we know, adding a dash of creativity, and mixing it all together in a clever way to build what once seemed impossible.

An Angle Bisector is a ray that splits the angle into two congruent angles.

Build It

Use the Geometer's Toolbox to construct the angle bisector of the given angle. The activity will indicate if you are successful. Hint an angle bisector is a ray. Use the ray tool to complete your build.

The Solution

Construct a circle centered at A
Mark the points where this circle intersects the rays of your angle, AB and AC. Call these points D and E
Using your previous constructions, construct the perpendicular bisector of DE
Call the intersection of your circles in the perpendicular bisector construction F and G where F is on the interior of ∠CAB
Create a ray starting at A and passing through F
You've created the angle bisector of ∠CAB!

Why it Works

Once again we need a circle, any circle. The actual size of the circle isn't important - what we care about is the properties of this character. The circle centered at A creates congruent segments AD and AE. When we connect points D and E we've formed one of our special and powerful characters - an Isosceles triangle ΔADE.

Leaning on the symmetry of the Isosceles triangle, we are now able to bisect our angle, essentially "folding" our triangle down this line of symmetry and cutting our angle in half too!