Axiomatic System and Geometry 2018 2019 9292020 10

Axiomatic System and Geometry 2018 -2019 9/29/2020 10: 45 PM

Some Basic Information of Geometry with Figures: - 9/29/2020 10: 45 PM

History of Geometry: Epochs in the development of geometry, from Egypt a knowledge of geometry was transferred to Greece, whence it spread to other countries. Hence we have the following principal epochs in the development of geometry; _ 1 - Egyptian 3000 B. C. - 1500 B. C. 2 - Greek 600 B. C. - 100 B. C. 3 - Indian 500 A. D. – 1100 A. D. 4 - Arab 800 A. D- 1200 A. D. 5 - European 1200 A. D. B. C. = Before century A. D. = Anno Domini 9/29/2020 10: 45 PM

Euclid’s Axioms system definitions: Euclid was put 23 definitions 1) A point is that which has no part. 2)A line has only length. 3) The extremities of a line are points. 4) A straight line is a line, which lies evenly with the points on itself. Or a straight line is that which lies evenly between its extreme points. 5) A surface is that which has length and width only. 6) The extremities of a surface are lines. 9/29/2020 10: 45 PM

Euclid’s Axioms system definitions 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 where lines containing the angles are straight the angle is called rectilinear. 10) And when a straight-line setup on a straightline makes adjacent angles equal to one another, each of the equal angles is a right-angle, and the straight-line standing on the other side is called a perpendicular to that on which it stands. 9/29/2020 10: 45 PM

Euclid’s Axioms system definitions 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 such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another, 9/29/2020 10: 45 PM

Euclid’s Axioms system definitions 16) And the point is called the center of the circle. 17) A diameter of the circle is any straight line drawn through the center and terminated in both directions by the circumference of the circle and such a straight line also bisects the circle. 18) A semicircle is the figure contained by the diameter and the circumference cut off by it. And the center of the semicircle is the same as that of the circle. 9/29/2020 10: 45 PM

Euclid’s Axioms system definitions 19) rectilinear figures are those which are contained by 2 straight lines, trilateral figures being those contained by 3 straight lines , quadrilateral figures those contained by 4 straight lines, and multilateral figures those contained by more than 4 straight lines. 20) of trilateral figures, an equilateral triangle is that which has its 3 sides equal, an isosceles triangle that which has 2 of its sides alone equal, and a scalene triangle that which has its 3 sides unequal. 9/29/2020 10: 45 PM

Euclid’s Axioms system 21) 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 3 angles acute. 22) of quadrilateral figures, a square is that which is both equilateral and right angled, an oblong that which is right angled but not equilateral, a rhombus that which is equilateral but not right angled, and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right angled. And let quadrilaterals other than these be called trapezia (trapeziums). 9/29/2020 10: 45 PM

Euclid’s Axioms system 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. Axioms: - Axioms are statements, we take these statements to be true without proof, these are the basic ( rule of operation) of our system. 9/29/2020 10: 45 PM

First: common notation 1)things which are equal to the same thing are also equal to one another 2) if equals be added to equals, the results are equal. 3) if equals be subtracted from equals, the remainders are equal. 4) things which equivalent (coincide) with one another are equal to one another. 5) the whole (all) is greater than part. 9/29/2020 10: 45 PM

Second: postulates or axioms 1) to draw a straight line from any point to any point. 2) to produce 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) if a straight line falling on two straight lines makes the interior angles on the same side and the sum of interior angles is less than two right angles, the two straight lines if produced indefinitely, meet on the side angles. 9/29/2020 10: 45 PM

Proposition (1) To describe an equilateral triangle on a given finite straight line C A 9/29/2020 10: 45 PM B

Proposition(4) If two triangles have two sides of the one equal to two sides of the other, each to each, and have also the angles contained by those side equal, then shall their bases or third sides be equal , and the triangles shall be equal in area, and their remaining angles shall be equal, each to each, namely those to which the equal sides are opposite: that is to say the triangles shall be equal in all respects D 9/29/2020 10: 45 PM E A F B C

Proposition(6) if two angles of a triangle be equal to one another, then the sides also which or are opposites to the equal angles, shall be equal to one another C 4 3 1 A 9/29/2020 10: 45 PM 2 D B

Proposition(8) if two triangles have two sides equal to two sides respectively, and also have the base equal to the base, then they will also have equal the angles encompassed by the equal straight lines x 1 a 3 y z 4 2 9/29/2020 10: 45 PM h b c

Proposition(9) to cut a given rectilinear angle in half b d f a e c 9/29/2020 10: 45 PM

Proposition(10) to cut a given finite straight line in half C B A 9/29/2020 10: 45 PM D

Proposition(11) to draw a straight line at right angles to a given straight line from a given point on it F 2 A 9/29/2020 10: 45 PM D C 1 E B

Proposition(12) to draw a straight line perpendicular at a given infinite straight line from a given point which is not on it D C 2 9/29/2020 10: 45 PM A G 1 H E B

Proposition(15) if two straight lines cut one another, then they make the vertically opposite angles equal to one another. D 3 A 2 1 4 C 9/29/2020 10: 45 PM B

Proposition(16) for any triangle, when one of the sides is produced the external angle is greater than each of the internal and opposite angle F A 4 E 2 3 5 1 x D 9/29/2020 10: 45 PM C B

Proposition(27): if a straight line falling across two straight lines makes the alternate angles equal to one another, then the two straight lines will be parallel to one another e b 1 d f 9/29/2020 10: 45 PM a 2 c

Proposition(28): if a straight line falling across two straight lines makes the external angle equal to the internal and opposite angle on the same side, or makes the sum of the internal angles on the same side equal to two right angles, then the two straight lines will be parallel to one another e b 9/29/2020 10: 45 PM 2 f a 3 4 d 1 c

Proposition (29): A straight line falling across parallel straight lines makes the alternate angles equal to one another, the external angle equal to the internal and opposite angle, and the sum of the internal angles of the same side equal to two right angles e 3 b 1 4 2 d f 9/29/2020 10: 45 PM a c

Proposition (31): to draw a straight line parallel to a given straight line through a given point E A F 1 2 B 9/29/2020 10: 45 PM D C

Q/ the sum of the angles of a triangle is equal to two right angles A D 5 4 1 2 B 9/29/2020 10: 45 PM 3 C

Equivalent Axiom (E. A. ) E. A. with (1 -4) Euclid’s Axiom prove → E 5 A or prove E 5 A with (1 -4) Euclid’s Axiom → E. A. 9/29/2020 10: 45 PM

Some of Equivalent Axiom to Euclid’s fifth Axiom: 1) E 5 A 2)playfair’s axiom : through a given point can be drawn only one parallel to a given line. 3) if a straight line intersects one of two parallel lines, it will intersects the other also. 4) straight lines which are parallel to the same straight line are parallel to one another 9/29/2020 10: 45 PM

5) If in a quadrilateral three angles are right angles, then the fourth angle is also a right angle. 6) there exists a pair of similar triangles 7) the perpendicular distance is constant between two parallel lines. 8) the sum of the angles of a triangle equals two right angles. 9/29/2020 10: 45 PM

9) the sum of the angles of a quadrilateral equals four right angles 10) if two parallel lines falling a cross two other parallel lines the internal and external angles of one of them are equal, so the other external and internal angles are equal too. 9/29/2020 10: 45 PM

11)There exists a circle passing through any three non collinear points. 12) if the sum of the angles of any triangle equal to constant number then this constant number equal to two right angles. 9/29/2020 10: 45 PM

12) if the sum of the angles of any triangle equal to constant number then this constant number equal to two right angles. B β 2 D 1 α A 9/29/2020 10: 45 PM ε 2 ε 1 ε C

playfair’s axiom : through a given point can be drawn only one parallel to a given line. Is equivalent to Euclid’s fifth axiom case 1: if (1 -4) Euclid’s axiom is true with playfair’s axiom , then we prove the Euclid’s fifth axiom. E G H B α β D 9/29/2020 10: 45 PM F A γ C

case 2: if (1 -4) Euclid’s axiom is true with Euclid’s fifth axiom. , then we prove the playfair’s axiom. A α β C 9/29/2020 10: 45 PM B D

Hilbert’s Axioms system: Hilbert’s axioms are divided into 5 groups as following: Axioms of Connection, Order, Congruence, Continuity and Parallel. 1) Axioms of Connection : 1) for every 2 points A and B, there exists a unique line L that contains both of them. 2) there at least 2 points on any line. 3) there exist at least 3 points that do not all lie on a line. 4) if there are A, B, C points do not all lie on a line, there exist a unique plane X that contains all them. 9/29/2020 10: 45 PM

1) Axioms of Connection : 5) For each plane there exist a point on it. 6) if 2 points of a line L lie in a plane X, then all points of the line L lie on the plane X. 7) if 2 planes associate to a point, then there at least associate to an another point. 8) there exist at least 4 points do not lie on a plane. 9/29/2020 10: 45 PM

Example 1: use Axioms of Connection for Hilbert system to prove that ‘ every 2 different lines on plane are associates in a just point or not. Example 2: all straight line and point not on it, then form a unique plane. 9/29/2020 10: 45 PM

Example 3: if 2 planes are intersect, then their intersection is a line. Example 4: there exist at least six straight lines do not lie on a plane. 9/29/2020 10: 45 PM

2) Axioms of order 1) The points A, B, C are 3 distinct points of a line, then B is between A and C , and B is between C and A. 2) for 2 distinct points B and D, there are points A, C, and E such that B is between A , D and C is between B , D and D is between B , E 3) of any 3 distinct points on a line, there exists one and only one point between the other. 4) for every line L and points A, B and C not on L: i) if A and B are on the same side of L and B and C are on the same side of L, then A and C are on the same side of L. ii) if A and B are on opposite side of L and B and C are on opposite side of L, then A and C are on the same side of L. 9/29/2020 10: 45 PM

2) Axioms of order 5) Pasch’s theorem: - if triangle ABC is any triangle and L is any line intersects side AB in a point between A and B, then L also intersects either side AC or BC, if angle C is not incident with L. If not , then L does not intersect both AC and BC. B L A 9/29/2020 10: 45 PM C

3) Axioms of congruence 1) Let AB be any segment and A 1 be any point, on each line through A 1 can find 2 points B 1 and B 2 , such that A 1 is between B 1 and B 2 and A 1 B 1 AB, A 1 B 2 AB. 2) if AB CD and AB EF, then CD EF. Every segment is congruent to itself. 3) if B is a point on segment AC and B 1 is a point on segment A 1 C 1, and AB A 1 B 1 and BC B 1 C 1 then AC A 1 C 1. 4) given angle BAC and any ray A 1 B 1 there is a unique ray A 1 C 1 on a given side of A 1 B 1 such that angle BAC angle B 1 A 1 C 1. 9/29/2020 10: 45 PM

3) Axioms of congruence 5) if angle A angle B and angle A angle C, then angle B angle C. Every angle is congruent to itself. 6) if 2 sides and the included angle of one triangle are congruent respectively to 2 sides and the included angle of another triangle, then the two triangles are congruent. 9/29/2020 10: 45 PM

4) Axioms of continuity 1) Archimedes axiom: let A, A 1, B be 3 points on a segment, such that A 1 lies between A and B where A 1 is between A and A 2, A 2 is between A 1 and A 3, and so on, there exist a point Ax such that B is between A and Ax 2) Dedekind’s axiom: suppose that all points on line L are the union of two non-empty set ∑ 1 U∑ 2 such that no point of ∑ 1 is between 2 points of ∑ 2 and vice versa. Then there is a unique point O on L such that p 1*O*p 2 for any point p 1 ∑ 1 and p 2 ∑ 2. 9/29/2020 10: 45 PM

5) The Axioms on parallels Playfair’s postulate: for any line L and point p not on L there is exactly one line through p parallel to L. 9/29/2020 10: 45 PM

Radical axis • Radical axis of two intersecting circles is extension the joint chord of two intersecting circles at two direction of the circles Radical axis of two tangent circles, is joint tangent to the circles • Radical axis of two disjoint circles is perpendicular of the equal power of the point for two circles on the line between center of circles 9/29/2020 10: 45 PM

Basic properties of Inversion • The inverse of any line passing through center of inversion circle, is itself. • The inverse of any circle no passing through center of inversion circle, is itself. • The inverse of any circle passing through center of inversion circle, is a line no passing through center of inversion circle. • The inverse of any line no passing through center of inversion circle, is a circle passing through center of inversion circle. 9/29/2020 10: 45 PM

Cross Ratio: Any 4 distinct points A, B, C, and D determine a number {AB. CD} called the cross ratio of the points in this order, it is defined by Example: if A(2, 3), B(3, 4), C(5, 6) and D(8, 9) find the cross ratio for these points. H. W: if A(2, 3, 4), B(3, 5, 7), C(4, 7, 10) and D(0, -1, -2) find the cross ratio for these points. 9/29/2020 10: 45 PM