What should I learn first in elementary geometry?

Geometry is the study of spatial structure and properties. It is one of the most basic research contents in mathematics, which has the same important position as analysis and algebra, and has a very close relationship.

The earliest geometry belongs to plane geometry. Plane geometry is to study the geometric structure and measurement properties (area, length, angle) of straight lines and quadratic curves (that is, conic curves, that is, ellipses, hyperbolas and parabolas) on the plane.

Several famous theorems of plane geometry

1, Pythagorean Theorem (Pythagorean Theorem) 2, Projective Theorem (Euclid Theorem) 3, The three median lines of a triangle intersect at a point, and each median line is divided into two parts of 2: 1 by this point. 4. The lines connecting the centers of two sides of the quadrilateral intersect with the lines connecting the centers of two diagonal lines at one point. 5. The centers of the two triangles connecting the sides of the hexagon at regular intervals are 6. The perpendicular bisector of each side of a triangle intersects at a point. 7. The three lines of a triangle intersect at one point. 8. Let the outer center of the triangle ABC be O and the vertical center be H, draw a vertical line from O to BC, and let the vertical foot be L, then AH=2OL 9. The outer center, vertical center and center of gravity of a triangle are on the same straight line (Euler line). 10, (nine-point circle or Euler circle or Fellbach circle) triangle, the center of three sides, drawn from each vertex to the vertical foot on the opposite side, connects the center of the vertical foot with the midpoint of each vertex, and these nine points are on the same circle. 1 1, euler theorem: the outer center of the triangle, the center of gravity, the center of the nine-point circle and the vertical center are located on the same straight line (Euler line) in turn. 12, coolidge's theorem: (A nine-point circle with a quadrilateral in it) There are four points on the circumference, and any three of them are triangles, and the nine-point centers of these four triangles are on the same circumference. 13, the bisectors of the three inner angles of the (inner) triangle intersect at one point, and the radius formula of the inscribed circle is: r=(s-a)(s-b)(s-c)s, where s is the half circumference of the triangle,14; A bisector of the inner angle of a (near-center) triangle intersects with the bisectors of the outer angles of the other two vertices at a point,15; Mean value theorem: (babs's theorem) If the midpoint of the side BC of the triangle ABC is p, there will be AB2+AC2=2(AP2+BP2) 16. Stuart's theorem: if P divides the side BC of the triangle ABCD into m:n, there will be n× AB2+m× AC2 = (m+n) AP2+. The straight line connecting the midpoint M of Ab and the diagonal intersection E is perpendicular to CD 18. Apollonius theorem: the point P with a constant ratio of m:n (the value is not 1) is located on a fixed circle that divides the line segment AB into the inner point C and the outer point D of m:n as the two ends of the diameter. Ptolemy theorem: set a quadrilateral AB. Then there is AB×CD+AD×BC=AC×BD 20. With the sides BC, CA and AB of any triangle △ABC as the base, the isosceles △BDC, △CEA and △AFB with the base angle of 30 degrees are respectively made outward, then △DEF is a regular triangle, 2 1 and Elk's theorem1:. Elkos Theorem 2: If △ABC, △DEF and △GHI are all regular triangles, then the triangle composed of the center of gravity △ADG, △BEH and △CFI is a regular triangle. 23. Menelaus Theorem: Let the intersections of three sides BC, CA and AB of △ABC or their extension lines and a straight line that does not pass through any of their vertices be P, Q and R respectively, then BPPC× CQQA× ARBB = 1 24, the inverse theorem of Menelaus Theorem: (omitted) 25, the application of Menelaus Theorem. 26. Menelaus Theorem Application Theorem 2: If the three vertices A, B and C of any △ABC are tangent to its circumscribed circle and intersect with the extension lines of BC, CA and AB at points P, Q and R respectively, then the three points P, Q and R are collinear. 27. Seva Theorem: Let the three vertices A, B and C of △ABC not be on the sides of the triangle or on their sides. If a straight line intersects with sides BC, CA, AB or their extension lines at points P, Q, R, then BPPC×CQQA×ARRB()= 1. 28. Application Theorem of Seville Theorem: Let the intersections of straight lines parallel to side BC and two sides AB and AC BE D and E respectively, and let Be and CD intersect at S, then AS will cross the center of BC M 29, the inverse theorem of Seville Theorem: (omitted) 30, the application theorem of the inverse theorem of Seville Theorem 1: The three median lines of the triangle intersect at a point 3 1. 32. simonson's Theorem: Take any point P on the circumscribed circle of △ABC as the vertical line of three sides BC, CA and AB or their extension lines, and let its vertical foot be D, E and R, then D, E and R are collinear (this straight line is called simonson line) 33. Inverse theorem of simonson's theorem: (omitted) 34. Steiner theorem: set. 35. The application theorem of Steiner's theorem: the symmetry point of a point P on the circumscribed circle of △ABC about the sides BC, CA and AB is on a straight line (parallel to Simpson's line) with the vertical center H of △ABC. This straight line is called the mirror image of point P about △ABC. 36. Blanchot and Tengxia Theorem: Let three points on the circumscribed circle of △ABC be P, Q and R, then the necessary and sufficient conditions for P, Q and R to intersect at one point are: arc AP+ arc BQ+ arc CR=0(mod2∏). 37, Braun Day and Tengxia Theorem Inference 65438+. Then the Simpson lines of points A, B and C of △PQR intersect at the same point as before. 38. Inference 2 of Polanje and Tengxia Theorem: In inference 1, the intersection of three Simpson lines is the midpoint of the connecting line between the vertical center of the triangle formed by three points A, B, C, P, Q and R and the vertical center of the triangle formed by the other three points. 39. Braunge and Tengxia Theorem Inference 3: Investigate the Simpson line of point P on the circumscribed circle of △ABC. If QR is the chord of a circumscribed ballpoint pen perpendicular to this Simpson line, then the Simpson line about △ABC intersects at point 40 at three points: P, Q and R.. Braunge and Tengxia Theorem Inference 4: From the Vertex of △ABC to the Edge of BC and BC. And let the midpoints of BC, CA and AB be L, M and N respectively, then six points of D, E, F, L, M and N are on the same circle, then the points of L, M and N intersect at one point about △ABC and Simpson line. 4 1, Theorem on Seymour Line1:△ The two endpoints P and Q of the circumscribed circle of ABC are perpendicular to each other with respect to Seymour Line of the triangle, and their intersection points are on the nine-point circle. 42. Theorem 2 on Simpson Line (Peace Theorem): There are four points on a circle, any three of which are triangles, and then the remaining points are Simpson lines about triangles, and these Simpson lines intersect at one point. 43. Carnot Theorem: Through a point P of the circumscribed circle of △ABC, straight lines PD, PE and PF with the same direction and the same angle as the three sides BC, CA and AB of △ABC are introduced, and the intersections with the three sides are D, E and F, respectively, so that the three points of D, E and F are collinear. 44. aubert's Theorem: Draw three parallel lines from the three vertices of △ABC, and their intersections with the circumscribed circle of △ABC are L, M and N respectively. If a point p is taken from the circumscribed circle of △ABC, the intersections of PL, PM and PN with BC, CA, AB or their extension lines are D, E and F respectively. Then d, e and f are collinear. 45. Qing Palace Theorem: Let P and Q be two points on the circumscribed circle of △ABC, which are different from A, B and C, and the symmetry points of point P about BC, CA and AB are U, V and W respectively. At this time, the intersection points or extension lines of flexion, flexion and edge BC, CA and AB are D, CA and AB, respectively. Then d, e and f are collinear 46. Other theorems: Let P and Q be a pair of antipodes about the circumscribed circle of △ABC, and the symmetry points of point P about BC, CA and AB are U, V and W respectively. At this time, the intersections of Ruoqu,, and sides BC, CA, AB or their extension lines are respectively ED, E, F. (Anti-point: P and Q are the two points of the radius OC of the circle O and its extension line respectively. If OC2=OQ×OP, then the two points of P and Q are called antipodes relative to the circle o)47. Langerhans Theorem: There is a point A1B1C1D14 on the same circle, where any three points are triangles. 48. Nine-point Circle Theorem: The midpoints of three sides of a triangle, the vertical feet of three heights and three Euler points [the midpoints of three line segments obtained by connecting the vertices of the triangle with the vertical center] are nine points of a circle [this circle is usually called [nine-point circle], or Euler circle or Feuerbach circle. 49. There are n points on a circle, and the center of gravity of any n- 1 point is on this circle. 50. Cantor Theorem 1: There are n points on a circle, and the vertical lines drawn from the center of gravity of any n-2 points to the other two points are common. 5 1, Cantor Theorem 2: If there are four points A, B, C and D and two points M and N on a circle, then the intersection points △BCD, △CDA, △DAB and △ABC of the two Simpsons of each of these four triangles are on the same straight line. This straight line is called the Cantor line about point M and point N of quadrilateral ABCD. 52. Cantor Theorem 3: If there are four points A, B, C and D and three points M, N and L on a circle, then the Cantor line of quadrilateral ABCD at M and N, the Cantor line of quadrilateral ABCD at L and N intersect at one point. This point is called cantor point of m, n and l about quadrilateral ABCD. 53. Cantor Theorem 4: If there are five points A, B, C, D and E and three points M, N and L on a circle, then the three points M, N and L are on a straight line with respect to each Cantor point in the quadrilateral BCDE, CDEA, DEAB and EABC. This line is called Cantor line 54 of M, N and L about pentagons A, B, C, D and E. Fairbach theorem: the nine-point circle of a triangle is tangent to the inscribed circle and the circumscribed circle. 55. Morley Theorem: If three internal angles of a triangle are divided into three equal parts and two bisectors near one side get an intersection point, then such three intersections can form a regular triangle. This triangle is usually called Molly's regular triangle. 56. Newton's theorem 1: the midpoint of the line segment connected by the intersection of the extension lines of two opposite sides of a quadrilateral is collinear with the midpoint of two diagonals. This straight line is called Newton line of this quadrilateral. 57. Newton's Theorem 2: The midpoint of the two diagonal lines of the circumscribed quadrangle is collinear with the center of the circle. 58. Gilad Girard Desargues Theorem 1: There are two triangles △ABC and△ △DEF on the plane. Let the connecting lines of their corresponding vertices (A and D, B and E, C and F) intersect at one point. At this time, if the corresponding edges or their extension lines intersect, the three intersection points are collinear. 59. Gilad Girard Desargues Theorem 2: There are two triangles △ABC and △DEF in different planes. Let the connecting lines of their corresponding vertices (A and D, B and E, C and F) intersect at one point. At this time, if the corresponding edges or their extension lines intersect, the three intersection points are collinear. 60. Bryansson Theorem: If vertices A and D, B and E, C and F of hexagonal ABCDEF tangent to a circle are connected, then these three lines are common. 6 1, Basija theorem: the opposite side AB of inscribed hexagon ABCDEF is collinear with the intersection points of DE, BC and EF, CD and FA (or extension line).