Euclidean Geometry

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Euclidean geometry two-dimensional geometry based on Euclid's five postulates (self-evident propositions), 23 definitions and 5 supplemental axioms.

The five Euclid postulates are the fundamental assumptions on which Euclid's Elements, a mathematical treatise written in ancient Greece around 300 BCE, is based. These postulates are as follows:

1. A straight line segment can be drawn between any two points.
2. Any straight line segment can be extended indefinitely in a straight line.
3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
4. All right angles are congruent (equal to each other).
5. If a straight line intersects two other straight lines, and the sum of the inner angles on one side is less than two right angles, then the two straight lines, if extended indefinitely, will meet on that side.

These postulates were used by Euclid to develop the entire mathematical system presented in his Elements. They were later modified and expanded upon by other mathematicians, but they remain a crucial starting point for the study of geometry and mathematical reasoning.

The five Euclid axioms, also known as postulates, are the fundamental assumptions on which Euclid's Elements, a mathematical treatise written in ancient Greece around 300 BCE, is based. These axioms are as follows:

1. A straight line may be drawn from any two points.
2. A finite straight line may be extended indefinitely in a straight line.
3. A circle may be described with any center and any radius.
4. All right angles are equal to each other.
5. If a straight line intersects two other straight lines in such a way that the sum of the inner angles on one side is less than two right angles, then the two straight lines, if extended indefinitely, will intersect on that side where the angles are less than two right angles.

These axioms were used by Euclid to construct the entire mathematical system presented in his Elements. They were later modified and expanded upon by other mathematicians, but they remain a crucial starting point for the study of geometry and mathematical reasoning. 

The 23 Euclid definitions are the foundational definitions used in Euclid's Elements, a mathematical treatise written in ancient Greece around 300 BCE. These definitions are as follows:

1.           A point is that which has no part.

2.           A line is a length without breadth.

3.           The ends of a line are points.

4.           A straight line is a line which lies evenly with the points on itself.

5.           A surface is that which has length and breadth only.

6.           The edges of a surface are lines.

7.           A plane surface is a surface which lies evenly with the straight lines on itself.

8.           A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line.

9.           And when the lines containing the angle are straight, the angle is called rectilinear.

10.        When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.

11.        An obtuse angle is an angle greater than a right angle.

12.        An acute angle is an angle less than a right angle.

13.        A boundary is that which is an extremity of anything.

14.        A figure is that which is contained by any boundary or boundaries.

15.        A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference are equal to one another.

16.        And this point is called the center of the circle.

17.        A diameter of a circle is a straight line drawn through the center and terminated in both directions by the circumference of the circle.

18.        A semicircle is the figure contained by the diameter and the circumference cut off by it.

19.        A segment of a circle is the figure contained by a straight line and a circumference of a circle.

20.        Rectilinear figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines.

21.        Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal.

22.        Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute-angled triangle that which has its three angles acute.

23.        Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.