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Euclid's axioms and the structure of proof
Points, lines, the five postulates, and why geometry was the first axiomatic science.
Euclid began with five postulates — two points make a line, a line extends, a circle has any centre and radius, right angles are equal, and the parallel postulate — and derived hundreds of theorems from them by proof alone. Picture it: every construction below is a drawing made with a straightedge and compass, the only tools the postulates allow. Think it: the same postulates with the fifth one changed give hyperbolic and spherical geometry, which is how mathematics learned that axioms are choices.
Eżempju maħdum: triangle 5 5 5
Pass b'pass
- a = 5,\ b = 5,\ c = 5
Three sides (SSS). Check the triangle inequality: each side is less than the sum of the other two. ✓
- P = a + b + c = 15
Perimeter.
- s = \tfrac{P}{2} = \frac{15}{2},\quad A = \sqrt{s(s-a)(s-b)(s-c)} = \frac{25 \sqrt{3}}{4} \approx 10.825
Heron's formula for the area.
- \cos A = \frac{b^2 + c^2 - a^2}{2bc} = \frac{1}{2} \Rightarrow A \approx 60.000^\circ
Law of cosines for angle A (opposite side a).
- \cos B = \frac{a^2 + c^2 - b^2}{2ac} = \frac{1}{2} \Rightarrow B \approx 60.000^\circ
Law of cosines for angle B (opposite side b).
- \cos C = \frac{a^2 + b^2 - c^2}{2ab} = \frac{1}{2} \Rightarrow C \approx 60.000^\circ
Law of cosines for angle C (opposite side c).
- A + B + C = 180.0^\circ
The angles add to 180° — a acute, equilateral triangle.
Jiżvelaw it-tweġiba
Symbols used here
The non-negative number whose square (n-th power) is x.
Ratios of sides in a right triangle; coordinates on the unit circle.
Logical connectives.
1/360 of a full turn. 180° = π radians.
Equal to the precision shown, not exactly.
Ratio of a circle's circumference to its diameter, 3.14159…
The usual name for an angle.
The angle at B between BA and BC; the triangle with those vertices.
Never meet; meet at 90°; identical shape and size; same shape.
How to: Euclid's axioms and the structure of proof
- State what is given and what is to be proved.
- Draw the figure and label every point.
- Write each step with the postulate, definition or earlier theorem that justifies it.
- Stop when the statement to be proved appears as a line — that is the ∎.
Questions people ask
Why is the parallel postulate special?
It cannot be proved from the other four: replacing it gives consistent non-Euclidean geometries. Two thousand years of attempts to prove it ended with that discovery.
Why does every triangle have angles adding to 180°?
Draw a line through one vertex parallel to the opposite side: the two alternate angles equal the other two angles of the triangle, and the three angles at the vertex lie on a straight line. That is Euclid's proof, and it is why the fact holds only on a flat plane.
When do I use the law of sines versus the law of cosines?
Cosines when you know three sides, or two sides and the angle between them. Sines when you know an angle and the side opposite it, plus one more piece.
What is the difference between area and perimeter?
Perimeter is the length of the boundary (one dimension, measured in metres); area is the amount of surface inside (two dimensions, square metres). Doubling every side doubles the perimeter but quadruples the area.
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TrianglesPythagorean theoremCirclesPolygonsSolidsCoordinate geometryCongruent and similar trianglesCircle theoremsTransformations: translation, rotation, reflection, dilationSolid geometry: prisms, pyramids, spheres