Euclid's five theorems are: any two points can be connected by a straight line; Any line segment can be infinitely extended into a straight line; Given any line segment, you can make a circle with an endpoint as the center and a line segment as the radius; All right angles are congruent; If both straight lines intersect with the third line, and the sum of internal angles on the same side is less than the sum of two right angles, then the two straight lines must intersect at that side.

Euclid's geometric theorem refers to the geometry constructed according to Euclid's Elements of Geometry. Euclidean geometry sometimes refers to the geometry on a plane, that is, plane geometry. Euclidean geometry in three-dimensional space is usually called solid geometry. Before Euclid, the ancient Greeks had accumulated a lot of geometric knowledge and began to use logical reasoning to prove the conclusions of some geometric propositions.

Euclid sorted out many early theorems that were unrelated and not strictly proved, and wrote the book "Elements of Geometry", which marked the establishment of Euclid's geometry.

1 axioms are equal to each other.

Two equals plus equals.

Three equals negative equals.

4 things that are completely coincident are equal.

5 the assumption that the whole is greater than the parts.

From the non-Euclidean geometry founded by Lobachevsky, we can draw an extremely important and universal conclusion: a set of logically contradictory assumptions may provide a geometry.

1, any two points can be connected by a straight line.

2. Any line segment can be infinitely extended into a straight line.

3. Given any line segment, one of its endpoints can be used as the center of the circle, and the line segment can be used as a radius to make a circle.

4. All right angles are congruent.

5. If two straight lines intersect with the third straight line, and the sum of internal angles on the same side is less than two right angles, then the two straight lines must intersect at that side. The fifth postulate, called parallel axiom, leads to one of the biggest mathematical and philosophical problems in the Millennium.

Later generations proved that it is equivalent to the following two propositions:

1。 The sum of the internal angles of a triangle is equal to two right angles;

2. Through points that are not on a straight line, only one straight line does not intersect with this straight line.