September 1, the beginning of school, we saw a teacher about 50 years old, a look is called math, hands still holding a math book which, this teacher left me a lot of images, he was in the classroom, we listen to him talk, simply do not understand what he was saying, math class with is harsh ah, noon did not write the homework can not go to dinner, all 11:45 are not to go to dinner, and then he looked at his watch, oops, 11:30 are over before calling us to go to dinner, we have been hungry. watch, oops, 11:30 are over, only called us to go to dinner, we have been hungry, we ran to the cafeteria, a look at the meal has been gone.

We hate this teacher, so that we do not eat, and even more hateful is that in the afternoon, he said, we only have 4 math classes a week, and now there are actually 5 more math classes.

This math teacher is not without merit, there are, for example, he does not hit people, do not want to hit people like our former math teacher, is always called in class we are not good at math.

In fact, this math teacher is also very good. His name is Mr. Xie.

Math handbook content: math knowledge

I. Plane Geometry

1. (i) Nine-point circle theorem: the midpoints of the three sides of a triangle, the vertical feet of the three heights, the center of the vertical center and the midpoint of the line between the vertices of the nine points **** circle. (Nine-point circle is also known as Euler's circle, Feuerbach's circle)

(ii) Feuerbach's Theorem: The nine-point circle of a triangle is tangent to its internal tangent circle and the three collateral tangent circles.

(iii) Coolidge-Oshigami theorem: three of the four points on the circumference of the nine-point circle (arbitrarily taken) are triangulated, and all four triangles have the nine-point circle centered on the *** circle.

2. Simson (Simson) Theorem: over the triangle on the outside of the circle is different from any point on the triangle vertex for the three sides or its extension line on the vertical line, then the three perpendiculars*** line. (This line is often called Simson line)

3. Butterfly Theorem: Let M be the midpoint of the chord PQ in the circle, and make the chords AB and CD through M. Let AD and BC intersect PQ at point X and Y, then M is the midpoint of XY. (With a picture la la la la ~)

4. You know that there are Newton's three laws in physics, but you don't know that there are also Newton's three theorems in plane geometry (stop, of course, it is the same Newton), when I first learned about it, I simply worshipped it~

Newton's Theorem 1: The midpoints of the three diagonals in a perfect quadrilateral are **** lines.

Newton's Theorem 2: the midpoints of the two diagonals of a circle tangent to a quadrilateral, and the center of the circle, three points of the **** line. Extension: the trajectory of the center of a centered conic curve tangent to the four sides of a perfect quadrilateral is a straight line, a line **** by the midpoints of the three diagonals of the perfect quadrilateral.

Newton's Theorem 3: The intersections of the diagonals of the externally tangent quadrilateral of a circle coincide with the intersections of the diagonals of the quadrilateral with the tangent point as its vertex. (Quadrilateral *** point)

5. Pascal's Theorem: the intersection of the conic curve inside a hexagon its three opposite sides *** line, dyadic to Briensand's Theorem, is a generalization of Pappus's Theorem. (As for the latter two is what, poke in to see on the good, back then also just know what and did not use ~)

6. Root center theorem: three two different heart of the circle, the formation of the three root axis, then either the three axes two parallel, or three axes are completely overlap, otherwise three axes two intersect, that is, at this time, three axes must be intersected at a point (three line **** point), which is known as the center of the root of the three circles. (The root axis is the set of points on the two circles and other powers, is a perpendicular to the line of the center of the line, special circumstances: if the two circles intersect, the root axis is connected to the two public **** point of the straight line; if the two circles are tangent, the root axis is the common tangent over the tangent point;)

7. five **** circle: (specific tracing back to the roots of the search for the Miquel (Miquel) Theorem) (will not be able to prove that the children or not film first, hurry! Read more, or still naive ~~)

8. chicken claw theorem (I also want to know if there is a better name ah pro ~): set △ABC inside the heart of the I, ∠A inside the side of the center of the J, the extension of the line of AI intersection of triangles connected to the outside of the circle in the K, then the KI = KJ = KB = KC. (Note that the shape of the red line)

9. Napolé; 9. ;eacute;on) theorem (said to be marching and fighting to prove, also powerful): to any triangle three sides respectively to the outside as equilateral triangles, and then join the center of these three positive triangles formed by . The triangle must be equilateral.

This theorem can be equivalently described as follows: if you make isosceles triangles with base angles of 60° to the outside of the shape with each side of any triangle as a base, their centers form an equilateral triangle.

Some derivations:

1) On quadrilaterals, an analogous theorem is the van Aubel theorem.

2) Napoleon's theorem itself is a special case of the Pettenot-Iman-Douglas theorem.

3)The area of the inner Napoleon triangle is greater than or equal to 0 giving the Exenbeek inequality.

10. Morley's Theorem: If the three interior angles of a triangle are divided in thirds, and the two trisectors close to a side intersect to obtain an intersection, then such three intersections can form a square triangle. This triangle is often called a Morley square triangle. (Off-topic: I heard a high school classmate say, a teacher in the outside class to the innocent boys paper: and like the girl paper that casually draw a triangle, if its angle trisectors intersection happens to be a positive triangle, it proves that the love for her is sincere. I to that high school classmates immediately said, this is red fruit deception ah ~ now finally understand why they are still woof woof woof ~ ~ ~)

11. Euler Line Theorem (thanks to the comment section of the knowledge of the friends reminded ~): the outer center of any triangle, the center of gravity, the center of gravity, the center of pendant center, nine-point circle circle center, in turn, is located on the same straight line. (This line is called the Euler line of the triangle, and the distance from the center of gravity to the center of gravity is equal to half the distance from the center of gravity to the center of the pendant center)

12. Sawa-yama Theorem (thanks to the comment section of the know-how to remind ~): Circle P and the inner quadrilateral ABCD diagonal of circle O, AC, BD, cut in E, F, at the same time, and the circle O, tangent to E, F and the inner heart of △ ABD, △ ACD, I, I'**** line ( Four points *** line).

Plane geometry chapter to be continued

II. Elementary Algebra chapter (although the conclusions of many of the inequalities are also beautiful, so if no one specifically asks me for it, I won't change this part. After all, in terms of interesting and incredible, constant equations will be more impressive)

1. Euler's formula: (out of the immense admiration and worship of the great god Euler and the unique appreciation of this formula, the answerer must put it out first, but it may be too familiar to you all~)

, from which there is a constant equation that is often called the so-called "God's formula", and which has a very high degree of accuracy.

2. (i) For any natural number n,

the value of n is a positive integer.

(ii) For any natural number n,

is divisible.

III. Combinatorial Math Part

1. For simple polyhedra. Let V be the number of vertices, E the number of prisms and F the number of faces.

For any planar graph, Euler's formula can be generalized as:, where C is the number of connected branches in the graph.

For non-planar graphs, Euler's formula can be generalized as follows: if a graph can be embedded in a manifold M, then:,is the Eulerian schematic number of this manifold, which is invariant under successive deformations of the manifold. The Eulerian eigenvalue of a single connected manifold (e.g., a sphere or a plane) is 2.

2. There are only five types of ortho-polyhedra: ortho-tetrahedron, ortho-hexahedron (ortho-square), ortho-octahedron, ortho-dodecahedron, and ortho-twentyhedron.

Fourth, mathematical analysis (I do not know whether the classification is reasonable orz)