Introduction to Euclids Geometry

INTRODUCTION TO EUCLID’S GEOMETRY -

NCERT TEXTBOOK -

5.2 EUCLID’S DEFINITIONS, AXIOMS AND POSTULATES -

Though Euclid defined a point, a line, and a plane, the definitions are not accepted by mathematicians. Therefore, these terms are now taken as undefined.

1. A point is that which has no part. It does not have any size, shape, or dimensions.

2. A line is a breadthless length.

3. A straight line is a line which lies evenly with the points on itself. The ends of a line are points.

4. A surface is which lies evenly with the points on itself. The edges of a surface are lines.

5. A plane surface is a surface which lies evely with the straight lines on itself.

6. Consider the three steps from solids to points (solids-surfaces-lines-points). In each step we lose one extension, also called a dimension. points (no dimension) combine to form lines, lines (one dimension) combine to form surfaces, and surfaces (two dimension) combine to form solids (three dimension).

Euclid assumed certain properties, which were not to be proved. These were the ‘obvious universal truths‘. Two types — postulates and axioms.

Postulates are assumptions specific for geometry. Common notions were axioms and applied throughout mathematics.

Euclid’s axioms -

1. Things which are equal to the same thing are equal to one another.

2. If equals are added to equals, the wholes / sum is equal

3. If equals are subtracted from equals, the remainders are equal.

4. Things which coincide with each other are equal to each other. If two things are identical (that is,they are the same), then they are equal. In other words, everything equals itself. It is the justification of the principle of superposition

5. The whole is always greater than the part.

6. Things which are double of the same things are equal to one another.

7. Things which are halves of the same things are equal to one another.

8. Given two distinct points, there is a unique line passing through them.

These axioms refer to magnitudes of some kind. Magnitudes of the same kind can be added and compared, but cannot be done for different magnitudes.

Euclid’s Postulates -

1. A straight line may be drawn from any point to any other point — at least one straight line passes through two distinct points. It is frequently assumed that it is a unique line joining two distinct points. It is self evident, so it is taken as an axiom (8).

2. A terminated line can be produced indefinitely — terminated line is a line segment. this postulate infers that both sides of a line segment can be extended on either sides to form a line.

3. A circle can be drawn with any centre and any radius.

4. All right angles are equal to one another.

5. If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles.

Theorem 5.1 - Two distinct lines cannot have more than one point in common.

Equivalent Versions of 5th postulate -

Playfair’s Axiom - For every line ’l’ and for every point ’P’ not lying on ’l’, there exists a unique line ’m’ passing through ’P’ and parallel to ’l’.

Two distinct intersecting lines cannot be parallel to the same line.

CYCLIC QUADRILATERAL

The properties of a cyclic quadrilateral help us to identify this figure easily and to solve questions based on it. Some of the properties of a cyclic quadrilateral are given below:

• In a cyclic quadrilateral, all the four vertices of the quadrilateral lie on the circumference of the circle.

• The four sides of the inscribed quadrilateral are the four chords of the circle.

• The measure of an exterior angle at a vertex is equal to the opposite interior angle.

• In a cyclic quadrilateral, p × q = sum of product of opposite sides, where p and q are the diagonals.

• The perpendicular bisectors are always concurrent.

• The perpendicular bisectors of the four sides of the cyclic quadrilateral meet at the center O.

• The sum of a pair of opposite angles is 180° (supplementary). Let ∠A, ∠B, ∠C, and ∠D be the four angles of an inscribed quadrilateral. Then, ∠A+∠C=180° and ∠B+∠D=180°.