Geometry is a branch of mathematics concerned with questions of shape, size, position, and space. A key figure was the Greek mathematician Euclid, whose book Elements systematized geometry in the 3rd century BC. Euclid defined fundamental terms like point and line, and postulated axioms and rules for reasoning about geometric concepts. Euclid's system of Euclidean geometry reigned for over 2000 years, until non-Euclidean geometries emerged in the 19th century challenging its assumptions about physical space.

1. By Arnav Singh (9 D)

2.  Geometry ( geo "earth", metron "measurement") is a
branch of mathematics concerned with questions of
shape, size, relative position of figures, and the
properties of space.
 A mathematician who works in the field of geometry is
called a geometer.
 Geometry was one of the two fields of pre-modern
mathematics, the other being the study of numbers.

3.  Euclidean Geometry is the study of geometry based on
definitions, undefined terms (point, line and plane) and the
assumptions of the mathematician Euclid (330 BC)
 Euclidean Geometry is the study of flat space
 Euclid's text Elements was the first systematic discussion of
geometry. While many of Euclid's findings had been
previously stated by earlier Greek mathematicians, Euclid is
credited with developing the first comprehensive deductive
system. Euclid's approach to geometry consisted of proving all
theorems from a finite number of postulates and axioms.
 The concepts in Euclid's geometry remained unchallenged
until the early 19th century. At that time, other forms of
geometry started to emerge, called non-Euclidean geometries.
It was no longer assumed that Euclid's geometry could be used
to describe all physical space.

4.  The Greek mathematicians of Euclid’s time thought of
geometry as an abstract model of the world they lived in.
 The notions of point, line , plane etc. were derived from
what was seen around them.
 Euclid summarised these notions as definitions. A few
are given below:
 A point is that of which there is no part.
 A line is a width less length.

5.  A straight line is one which lies evenly with the points
on itself.
 The extremities of lines are called points.
 A surface is that which has only length and breadth.
 The edges of surface are lines.
 A plane surface is a surface which lies evenly with
straight lines on itself.

6.  Axioms are assumptions used throughout mathematics
and not specifically linked to geometry. Here are some of
Euclid's axioms:
 Axiom 1: Things that are equal to the same thing are also
equal to one another (Transitive property of equality).
 Axiom 2: If equals are added to equals, then the wholes
are equal.
 Axiom 3: If equals are subtracted from equals, then the
remainders are equal.

7.  Axiom 4: Things that coincide with one another equal
one another (Reflexive Property).
 Axiom 5: The whole is greater than the part.
 Axiom 6: Things which are halves of the same things
are equal to one another
 Axiom 7: Things which are double of the same things
are equal to one another

8.  Postulates are assumptions specific to geometry.
 Let the following be postulated:
1. To draw a straight line from any point to any point.
2. To produce [extend] a finite straight line continuously in a
straight line.
3. To describe a circle with any centre and distance [radius].
4. That all right angles are equal to one another.
5. The parallel postulate: That, if a straight line falling on two
straight lines make the interior angles on the same side less
than two right angles, the two straight lines, if produced
indefinitely, meet on that side on which are the angles less
than the two right angles.

9.  THEOREM :-
Two distinct lines cannot have more than one point in
common
 PROOF :-
 Two lines ‘l’ and ‘m’ are given. We need to prove that
they have only one point in common Let us suppose
that the two lines intersects in two distinct points, say
P and Q

10.  That is two line passes through two distinct points P
and Q
 But this assumptions clashes with the axiom that only
one line can pass through two distinct points
 Therefore the assumption that two lines intersect in
two distinct points is wrong
 Therefore we conclude that two distinct lines cannot
have more than one point in common