All theorems of high school geometry, solid geometry commonly prove theorems high school.

If you have more, read the textbook or buy a booklet.

There's one kind of pamphlet in the bookstore.

There are six types:1Definitional Method.

2.Vertical method.

3.Projective theorem.

4.The three-perpendicular theorem.

5.Vector method.

6.Transformation method.

Extended Information: The Three-Perpendicular Theorem:

A straight line in a plane is perpendicular to the projection of a diagonal line passing through the plane in this plane, then it is also perpendicular to the diagonal line.

The inverse theorem of the three-perpendicular theorem: if a straight line in a plane is perpendicular to a diagonal line passing through the plane, then it is also perpendicular to the projection of the diagonal line in the plane.

1. The three-perpendicular theorem describes the vertical relationship between PO (oblique line), AO (projection), and A (straight line).

2. A and PO can intersect or be different.

3. The essence of the three-perpendicular line theorem is the determination theorem that an oblique line in the plane and a straight line in the plane are perpendicular.

Regarding the application of the three-perpendicular theorem, the key is to find the perpendicular line of the plane (datum). As for the projection, it is determined by the vertical foot, the oblique foot, and is therefore secondary. From the proof of the Sanshan Huiqiao perpendicular theorem is a program that proves a b:

One penalty, two shots, three certificates. i.e. the geometric model.

First, find the plane (datum) and the plane perpendicular;

Second, find the projective line, at which point a and b become a straight line and an oblique line on the plane;

Third, it is proved that the projective line is perpendicular to the line a, so that a is perpendicular to b.

1.In the theorem, the four scattered lines are all for the same plane;

2.The key to applying the theorem is to find"Datum"this frame of reference.

Vector proof of the three-perpendicular theorem.

1.It is known that PO and PA are the perpendicular line and oblique line of plane A, OA is the projection of PA in A, B belongs to A, and B is perpendicular to OA, and verify: B is perpendicular to PA

Proof: Because PO is perpendicular A, Po perpendicular B, and because OA perpendicular B vector Pa= (vector Po + vector OA).

So the vector pa multiplied by b = (vector po + vector oa) multiplied by b = (vector po multiplied by b) plus (vector oa multiplied by b) = o, so pa is vertical b.

2.It is known that PO and PA are the perpendicular line and oblique line of plane A, OA is the projection of PA in A, B belongs to A, and B is perpendicular to PA, and verify: B is perpendicular OA

Proof: Because PO is perpendicular A, Po Perpendicular B, and because Pa is perpendicular B, the vector OA = (vector Pa - Vector Po).

So the vector oa is multiplied by b = = (vector pa - vector po) multiplied by b = (vector pa multiplied by b) minus (vector po multiplied by b) = 0, so oa is vertical b.

3.It is known that the three planes OAB, OBC, and OAC intersect at a point O, and the angle AOB = angle BOC = angle COA = 60 degrees, and find the angle formed by the intersection line OA in the plane OBC.

The vector oa = (vector ob + vector ab), o is the heart, and because ab = bc = ca, the angle formed by oa to the plane obc is 30 degrees.

1.Parallel perpendicular relationships within the plane are not explained.

2.If the straight lines are parallel to a straight line in a plane and the lines are not in the plane, they are parallel.

3.If two intersecting lines in a plane are parallel to the other plane, then the two planes are parallel.

4.If a straight line is perpendicular to two intersecting lines in a plane, then the line is perpendicular to the plane.

5.If the line surface is perpendicular, the line is perpendicular to any straight line in this plane.

6.The line plane is perpendicular to the plane that passes through the line.

7.If a straight line is parallel to a plane, then the plane passing through the line intersects the plane and is parallel to the line.

The above can solve all the three-dimensional geometry problems we are doing now.

Whether it will make it depends on your creation-. - Don't be afraid when you see the problem of solid geometry, no matter how complicated it is, you can't come up with these few sentences......

In solid geometry, there are not many theorems used in plane geometry in junior high school, and the commonly used ones are: two straight lines parallel to the same straight line are parallel; A set of quadrilaterals that are parallel and equal to the opposite sides is a parallelogram; Parallelograms, opposite sides parallel to each other; the median line of the triangle, parallel to and equal to half of the third side; The middle line of the bottom edge of the isosceles triangle, the bisector of the corner of the top corner, and the height of the bottom edge are all three. Pythagorean theorem trick inverse theorem.

One. A straight line is parallel to a plane (judgment).

1.Determination theorem. If a line outside the plane is parallel to a line in the plane, then the line is parallel to the plane.

2.Application: Counter-evidence (proving that a straight line is not parallel to a plane).

Two. The plane is parallel to the plane (judgment).

1.Decision theorem: If two intersecting lines on a plane are parallel to the other, then the two planes are parallel.

2.Critical: Determine if the two planes have a common point.

3. Straight lines are parallel to the plane (property).

1.Properties: A straight line is parallel to a plane, then the intersection of any line with this plane is parallel to the line 2Application: Make a plane through this line and intersect a known plane, then the intersection line is parallel to the line.

4. The plane is parallel to the plane (property).

1.Properties: If two parallel planes intersect a third plane at the same time, then their intersecting lines are parallel.

2.Application: By making a plane that intersects with two parallel planes, the line is parallel.

Five: the straight line is perpendicular to the plane (theorem).

1.Decision theorem: A straight line is perpendicular to two intersecting lines in a plane, then the line is perpendicular to the plane.

2.Application: If a straight line is perpendicular to a plane, then the line is perpendicular to all the lines in that plane (Line Plane Perpendicular Line Perpendicular).

Six. The perpendicular plane to the plane (theorem).

1.If one plane passes the perpendicular line of the other, the two planes are perpendicular.

Or do a dihedral angle judgment).

2.Application: Find or make a perpendicular line in one of the planes, that is, realize the conversion of the vertical surface of the line plane and the vertical surface of the surface.

Seven. The plane is perpendicular to the plane (nature).

1.Property 1: Two perpendicular lines perpendicular to the same plane are parallel.

2.Property 2: If two planes are perpendicular, then a straight line perpendicular to the intersection line in one plane is perpendicular to the other.

3.Property 3: If the two planes are perpendicular to each other, then a straight line perpendicular to the second plane through a point in the first plane is in the first plane (property 3 is useless, you don't need to remember it).

The above is a collation of theorems and properties of solid geometry. It is a basic that must be remembered!

"Chaos Theory" (Butterfly Effect) is similar to (Artificial Effect) Butterfly Effect = Nuclear Bomb**: 1 + 1 = 2 1 + 2 = 3, 1 cubic space + 2 cubic meters of material gas, if it does not explode, it will burst. You guys can launch:

1*1=1 2*2=4 3*3=9。-1*-1=1 -2*-2=4 -3*-3=9。In high school chemistry there is an isotope addition reaction, the ratio of nitric acid to hydrochloric acid, and then aqua regia is obtained.

These equations can be used instead: in geometry (equivalence substitution). Columns:

1*1+1*1=2 This is the exact algorithm, if we add and multiply then then this equation is equal to 3, the same way: 2*2+1*2=6 If you add first then it is equal to 12, the change in these data is very amazing. Because nuclear bombs are man-made.

The flapping wings of a butterfly cause a tornado. It's all caused by animals, that is, "the power is unbalanced".