A map without ruler numbers

Erklärus showed Liri an old star map – drawn only with compasses and a set square, completely without measurements. “I know this! At home, positions were constructed exactly, not just measured.”

Erklärus: “That is called geometric construction – precise work with compasses and a set square. Let’s practise it together.”

✏️ The perpendicular bisector

The perpendicular bisector of a line segment is a line that divides the segment exactly in the middle and stands at right angles to it.

A B
Liri: “With the compasses I draw two circles, one around A, one around B, both with the same radius – bigger than half the segment.”
Erklärus: “Exactly. Where the two circles cross, you draw a straight line through – that is the perpendicular bisector.”

📐 The angle bisector

The angle bisector is a ray that divides an angle exactly into two equal parts.

Bisector
Erklärus: “Draw an arc from the vertex that crosses both arms. Then, from these two points, draw two more arcs of equal size – the point where they cross lies exactly on the angle bisector.”

🔺 Constructing triangles – the congruence conditions

A triangle can be constructed uniquely if you know exactly the right combination of measurements:

SSS
Three sides given
ASA
One side + both angles next to it
SAS
Two sides + the angle between them
SSA
Two sides + the angle opposite the longer side
Liri: “Why aren’t any three measurements enough?”
Erklärus: “Only these four combinations fix a triangle uniquely – with other combinations, several different triangles could come out.”
You know all three sides of a triangle. Which congruence condition applies?
🌌 Constructions in the universe

Old star charts and navigation instruments were often constructed purely with compasses and ruler – long before there were digital tools, astronomers found exact geometric positions in this way.

❌ Common mistake

For the perpendicular bisector both arcs must have the same radius – and the radius must be bigger than half the length of the segment, otherwise the circles don’t cross at all!

📝 Exercise 1
You know one side and the two angles next to it. Which congruence condition?
📝 Exercise 2
You know two sides and the angle between them. Which congruence condition?
📝 Exercise 3
A 60° angle is to be bisected. How big is each part?

A map, constructed exactly

Liri now constructed her own star map using only compasses and a set square – exactly, without a single wrong measurement.

Liri: “With compasses and a set square I can find positions without measuring a single number – just through the right construction.”
“Perpendicular bisector: two arcs of equal size around both end points. Angle bisector: arcs from the vertex. Congruence conditions: SSS, ASA, SAS, SSA.”
What you learned today
SSS · ASA · SAS · SSA
Only these four combinations fix a triangle uniquely
🌟

Chapter 44 mastered!

Liri can now construct exactly with compasses and a set square.

Erklärus’s notebook – page 44:
“Construction instead of measuring: perpendicular bisector, angle bisector, triangle constructions by SSS, ASA, SAS, SSA.”
← Back to overview 44 – Geometric Constructions