Plane Geometry Theorem, Fundamental Theorem of Plane Geometry

Menelaus' theorem (Meislaus line).

The three sides of the ABC are BC, CA, AB or their extension with a little A'、b'、c', then a'、b'、c'Collinearity is cb'/a'c·cb'/b'a·ac'/c'b=1

Ceva theorem (Ceva point).

The three sides of the ABC are BC, CA, AB or their extension with a little A'、b'、c', then AA'、bb'、cc'The sufficient and necessary condition for the three lines to be parallel or intersect at one point is ba'/a'c·cb'/ba'·ac'/c'b=1

Ptolemy's theorem.

The sum of the products of the two pairs of sides of a quadrilateral is equal to the product of its diagonals, and the sufficient and necessary condition is that the quadrilateral is inscribed in a circle.

Simson's theorem (Simson line).

The sufficient and necessary condition for the perpendicular collinear of the perpendicular line drawn from one point to the three sides of the triangle is that the point falls on the circumscribed circle of the triangle.

Pythagorean theorem: The sum of the squares of the two right-angled sides of a right-angled triangle is equal to the square of the hypotenuse.

2.The three midlines of the triangle intersect at one point, and each midline is divided into two parts of 2:1 by this point.

3.Let the outer center of the triangle ABC be O, the vertical center is H, and the perpendicular line from O to the BC side is L, then Ah=2ol.

4.The outer center of the triangle, the vertical center, and the center of gravity are in the same straight line.

5.Midline theorem: Let the midpoint of the side bc of the triangle abc be p, then there is 2ab+2ac=2(ap+bp).

6.Projective theorem: In a right-angled triangle, the height on the hypotenuse is the ratio of the projection of the two right-angled edges on the hypotenuse, and each right-angle is the mid-term of the projection on the hypotenuse and the proportion of the hypotenuse on the right-angled edge.

The basic theorems of solid geometry include the determination theorem that a straight line is parallel to a plane, the property theorem that a straight line is parallel to a plane, and the determination theorem that a plane is parallel to a plane.

If a line outside the plane is parallel to a line inside the plane, the line is parallel to the plane. If a straight line is parallel to a plane, and the plane passing through the line intersects this plane, then the line is parallel to the intersectional line.

If there are two intersecting lines in a plane that are parallel to the other, then the two planes are parallel. If two parallel planes intersect the third plane at the same time, then the resulting two intersecting lines are parallel. If a straight line is perpendicular to two intersecting lines in a plane, then the line is perpendicular to this plane.

If two straight lines are perpendicular to the same plane, then the two straight lines are parallel to each other. If one plane passes through one of the perpendicular lines of the other plane, the two planes are perpendicular to each other. If the two planes are perpendicular to each other, then the straight lines perpendicular to their intersection in one plane are perpendicular to the other plane.

Introduction to Solid Geometry:

Mathematically, solid geometry is generally used as a follow-up to plane geometry, and is the traditional name for the geometry of three-dimensional Euclidean space, because in fact this is roughly the space in which people live. Stereo mapping deals with the measurement of the volume of different shapes: cylinders, cones, cones, spheres, prisms, wedges, caps, etc.

The Pythagoreans dealt with spheres and regular polyhedra, but pyramids, prisms, cones, and cylinders were poorly known before the Platonic school set out to deal with them. Eudesses established their method of measurement, proving that the cone is one-third of the volume of a column of equal height and may have been the first to prove that the volume of a sphere is proportional to the cube of its radius.

The trinity of points, lines and planes is represented by column cone billiards. The distance is from the point, and the angle is made of lines. Perpendicular parallelism is the key point, proving that concepts need to be clarified.

Lines, lines, lines, surfaces, and faces, and three pairs are looped between each other. The equation is thought for the whole, and the consciousness is changed to make up for it. Before calculating, it must be proved that the number of graphic halls that have been moved out is drawn.

Three-dimensional geometric auxiliary lines, commonly used perpendicular lines and planes. The concept of projective pre-knowledge is very important, and it is the most critical for solving the problem.

In Euclid's "Geometric Primitive", Euclid gave 23 definitions, 5 postulators, and 5 axioms at the beginning. In fact, the commune he spoke of was what we later called the axioms, and his axioms were some methods used for calculations and proofs (e.g., axiom 1: equal to the equal quantity of the same quantity, axiom 5:).

The whole is greater than the part, etc.) The five axioms he gave were very closely related to geometry, which were later the axioms in our textbooks. They are:

Public Assumption 1: A straight line can be drawn from any point to any other point.

Public Assumption 2: A finite line segment can be extended.

Public Hypothesis 3: Draw a circle with any point as the center and any distance.

Public Hypothesis 4: All right angles are equal to each other.

Public Hypothesis 5: A straight line in the same plane intersects two other straight lines, and if the sum of the two internal angles on one side is less than the sum of two right angles, then the two straight lines intersect on this side after an infinite extension.

In these five axioms, Euclid did not naively assume the existence and compatibility of definitions. Aristotle pointed out that the first three hypotheses speak of the possibility of constructing lines and circles, so he is a statement of the immanence of two things. In fact, Euclid used this construction to prove many propositions.

The fifth public premise is very wordy, not as concise and easy to understand as the first four. What is declared is not something that exists, but something that Euclid himself thinks. That's enough to say that he's genius.

For about 2,100 years from Euclid to 1800, when Euclid proposed this axiom, although people did not doubt the correctness of the whole system, they were still concerned about this fifth axiom. Many mathematicians tried to remove this hypothesis from this system, but after several efforts to no avail, they could not push it from other public assumptions to the fifth hypothesis.

At the same time, mathematicians have also noticed that this postulate is both a discussion of the concept of parallelism (hence the axiom of parallelism) and a discussion of the sum of the internal angles of a triangle (i.e., the sum of internal angles). Gauss understood this very well, and he believed that the geometry of material space in the form of Euclidean geometry, and in 1799 he said that in a letter to his friend he believed that the parallel kilometer could not be deduced from other public assumptions, and he began to seriously work on the development of a new geometry that could be applied. In 1813, he developed his geometry, which was first called anti-Euclidean geometry, then starry geometry, and finally non-Euclidean geometry.

In his geometry, the inner angles of the triangle can rise to more than 180 degrees. Of course, Gauss is not alone in getting such geometry, there are three people in history. One was his partner, and the other was discovered independently by the son of Gauss's friend.

One of the interesting problems is that in non-Euclidean geometry there can be an infinite number of parallel lines that are a little beyond the old straight line.