Euclidean geometry

Euclidean geometry is the study of the figures (drawings) obeying axioms that posed Euclide in its work Elements. It is geometry such as it is taught to the college. It often too is called plane geometry when the figures are traced on a plane surface, or solid geometry when volumes are considered.

Geometry of Elements of Euclide only use regulate and it compass although knowledge of its time included also the approximate constructions called by neusis.
The rule makes it possible to trace right features ( right-hand sides, segments of right-hand side), and the compass makes it possible to bring back distances (and incidentally to trace circles). One can note here that the rule is not graduated; one is not interested in the distance like quantity of centimetres but like nonnumerical size.
The drawing is its own standard; one can as say as the properties of the drawing do not depend on its scale.
Also let us note that the reading of the figure in the Euclidean geometry is vital and gives information that the text does not give.

Synopsis 1 Geometrical objects 2 Postulates of Euclide 3 Constructions with the rule and the compass 4 Metric 5 Geometrical figures 6 See too

Geometrical objects

Geometry, like any science of abstraction, defines objects (attention, these definitions are naive):

• not : one can colour the point like a pinhead, it is in fact infinitely small ; one represents it by a cross made with the pencil (the intersection of two secant lines is a point); one in general names the points by a capital Roman letter, like With, B...

• right-hand side : it is a right feature null thickness, like a tended wire extrêment thin; the line is the shortest way between two points; one represents it by a feature of right pencil; one in general names the line by a tiny or Greek letter Roman capital ( D or?), but if one knows two points distinct from the right-hand side (for example With and B), one can name the line by putting the name of these two points between bracket (one speaks about the "right-hand side (AB)");

• the segment of right-hand side : it is a portion of right-hand side ranging between two points; one it represente by a feature right delimited by two small features perpendicular at the ends; if With and B are the points of the ends, the segment is named by putting these names between hooks (one speaks about the "segment [AB]");

• the half-line : it is the portion of a line which is on a side of a point of the right-hand side; if With is the point of "departure" of the half-line and B another point of this half-line, then it is noted [AB);

• ring : it is the whole of the points which are at the same distance of a particular point called center; it is represented by a round with a cross in the medium; the cross locates the center, the round locates the points of the circle; it is in general named by a capital Roman letter (for example C).

Postulates ofEuclide

These postulates are called "requests", maybe of the proposals a priori nonobvious but that one asks all the same to admit to be able to work. They are presented here in the version of the translation of Peyrard:

1. To lead one right-hand side of an unspecified point at an unspecified point.
2. To prolong indefinitely, according to its direction, a finished line.
3. Of an unspecified point, and with an unspecified interval, to describe a circumference of circle.
4. All the right angles are equal between them.
5. If a line, falling on two lines, fact interior angles on the same side smaller than two rights, these lines, prolonged ad infinitum, will meet side where the angles are smaller than two rights.

The last postulate is the known good 5^E postulate of Euclide. One can also formulate it like below:

• Are a line (d) and a point M located out of this line; it does not pass by M that only one line (of) parallel with D

Constructions with the rule and the compass

The principal construction of the geometry is undoubtedly the layout of the mediator of a segment.

The mediator of the segment [AB] is the line D who crosses perpendicularly [AB] in its medium I.

Theorem : The mediator of a segment is the whole of the points which are at equal distance of its ends.

Reciprocal theorem : The whole of the equidistant points of the ends of a segment is the mediator of this segment.

This is seen easily by noticing that if a point is considered M mediator, segments [AM] and [BM] are symmetrical compared to the mediator. One can also use it theorem of Pythagore in the right-angled triangles FRIEND and IMB and to show the equality of their hypotenuses.

Therefore, if one can build the mediator, one can thus determine the medium of a segment and to trace a perpendicular on a line.

For that, one opens the compass over a length higher than half the length of the segment, then, one traces two circles with this radius, one centered on With, the other on B (makes some, one can be satisfied to trace only arcs of circle). The intersection of the two circles consists of two points located at equal distance of With and of B, and which thus defines well the mediator.

Metric

Lengths, angles

Geometrical figures

The geometry studies objects made up of several segments

• triangle
• quadrilateral : trapezoid, parallelogram, rhombus, square
• others polygons

See too

• solid geometry
• vectorial geometry
• analytical geometry
• not-Euclidean geometry