Infographics: Introduction to Euclid's Geometry

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Introduction to Euclid's Geometry
Etymology
Geo (Earth) + Metron (To Measure) = 
Geometry means 'Earth Measurement'
Ancient Egypt
Used geometry for calculating granary 
volumes and building canals and 
pyramids
Indus Valley
Around 3000 BC, organised cities with 
parallel roads and drainage systems
Euclid's Elements
300 BC - Compiled all knowledge into 13 
books, the foundation of modern 
geometry
Fundamental Geometric Concepts
Three Dimensions of a Solid
A solid object has shape, size, and position, 
and can be moved about in space. Its edges 
are called surfaces, which separate different 
parts of space and have no thickness.
The edges of surfaces consist of curves or 
straight lines that end in points.
Undefined Terms
Mathematicians accept certain terms without 
definition to avoid endless chains of 
explanations:
Point - Has no part
Line - Breadthless length
Plane - Surface lying evenly
Euclid's Seven Axioms
0 1
Equality Transitivity
Things which are equal to the 
same thing are equal to one 
another
0 2
Addition Property
If equals are added to equals, 
the wholes are equal
0 3
Subtraction Property
If equals are subtracted from 
equals, the remainders are 
equal
0 4
Coincidence
Things which coincide with 
one another are equal
0 5
Whole vs Part
The whole is greater than the 
part
0 6
Doubles
Things which are double of 
the same things are equal
0 7
Halves
Things which are halves of the same things are equal
Euclid's Five Postulates
1
Line Through Two 
Points
A straight line may be 
drawn from any one 
point to any other point
2
Line Extension
A terminated line can be 
produced indefinitely
3
Circle 
Construction
A circle can be drawn 
with any centre and any 
radius
4
Right Angles
All right angles are equal to one another 
(90°)
5
Parallel Lines
If interior angles on one side sum to less 
than 180°, the lines meet on that side
Practical Examples
Example 1
Solve: y 2 10 = 13
Adding 10 to both sides:
y 2 10 + 10 = 13 + 10
y = 23
Uses Axiom 2
Example 2
If PT = QT and TR = TS, 
then prove PR = QS
Adding equations:
PT + TR = QT + TS
PR = QS
Uses Axiom 2
Weight Problem
Eric and David weigh the 
same. Both lose 5 kg.
Initial: x kg each
Final: (x 2 5) kg each
They still weigh the same!
Uses Axiom 3
Key Theorems
Unique Line Axiom
Given two distinct points, 
there is a unique line 
that passes through 
them. This is self-evident 
and forms the basis for 
much of geometry.
Two Lines, One 
Point
Two distinct lines cannot 
have more than one 
point in common. This 
prevents contradictions 
in geometric 
constructions.
Equilateral 
Triangle
An equilateral triangle 
can be constructed on 
any given line segment 
using circles with equal 
radii from both 
endpoints.
Summary of Key Definitions
Term Definition
Point That which has no part
Line A breadthless length whose ends are points
Straight Line A line which lies evenly with the points on itself
Surface That which has length and breadth only
Plane Surface A surface which lies evenly with straight lines on itself
Remember: Euclid's work 'Elements' consists of 13 books and contains 465 
propositions proved using axioms, postulates, and logical deduction. It remains the 
foundation of geometry even today!