Common conclusions on the usability of plane geometry in senior high school.

Know the famous theorem of 6 1 plane geometry, draw pictures and learn by yourself, and some need to be proved.

★ 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 line connecting the centers of two sides of the quadrilateral and the line connecting the centers of two diagonal lines intersect at one point.

5. The centers of gravity of two triangles formed by connecting the centers of the sides of a hexagon at intervals are coincident.

★6. The perpendicular bisector of each side of the triangle intersects at one point.

★7. Three vertical lines drawn from each vertex of a triangle to its opposite side intersect at one point.

8. Let the outer center of the triangle ABC be O and the vertical center be H. If a vertical line is drawn from O to BC and the vertical foot is not L, then AH=2OL.

9. The outer center, vertical center and center of gravity of a triangle are on the same straight 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, center of gravity, center of nine o'clock and vertical center of a triangle are located on the same straight line (Euler line) in turn.

12, Coolidge theorem: (The circle inscribed with the nine-point circle of a quadrilateral) has four points on its circumference, and any three of them are triangles, and the nine-point centers of these four triangles are on the same circumference. We call the circle passing through these four nine-point centers a nine-point circle inscribed with a quadrilateral.

★ 13. The formula of the radius of the inscribed circle when the three bisectors of the inner angle of the (inner) triangle intersect at one point:

S is half the circumference of a triangle.

★ 14. (near the center) A bisector of the inner corner of a triangle and a bisector of the outer corner intersect at one point at the other two vertices.

15, mean value theorem: (babs theorem) Let the midpoint of the side BC of the triangle ABC be p, then AB2+AC2=2(AP2+BP2).

16, Stuart Theorem: p Divide the side BC of the triangle ABC into m and n, then there is n×AB2+m×AC2=BC×(AP2+mn).

17, boromir and many theorems: when the diagonals of the quadrilateral ABCD inscribed in a circle are perpendicular to each other, the straight line connecting the midpoint m of AB and the diagonal intersection e is perpendicular to CD.

18, Apollonius Theorem: Point P (the value is not 1) with a constant ratio of m:n from two fixed points A and B is located on a fixed circle with the inner bisector C and the outer bisector D dividing the line segment AB into m:n at both ends of the diameter.

★ 19, Ptolemy theorem: If the quadrilateral ABCD is inscribed in a circle, there is AB×CD+AD×BC=AC×BD.

★20. Take the sides BC, CA and AB of any triangle ABC as the base, and make isosceles △BDC, △CEA and △AFB with the base angles outward of 30 degrees respectively, then △DEF is a regular triangle.

2 1, Elkos Theorem 1: If both △ABC and △DEF are regular triangles, the triangle formed by the barycentries of line segments AD, BE and CF is also regular triangles.

22. 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. Menelius Theorem: Let the intersections of three sides BC, CA, AB of △ABC or their extension lines with a straight line that does not pass through any of their vertices be p, q, r respectively, then BP/PC×CQ/QA×AR/RB= 1.

★24. The inverse theorem of Menelaus theorem: (omitted)

★25. Menelaus Theorem Application Theorem 1: Let the bisector of the outer corner of △ABC ∠A intersect with the bisector of Q and ∠C, AB intersect with the bisector of R and ∠B, and CA intersects with Q, then the straight lines of P, Q and R are * * *.

★26. Application Theorem 2 of Menelaus Theorem: If the three vertices A, B and C of any △ABC are tangents of 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 * * lines.

★27. Seva Theorem: Let △ABC's three straight lines formed by point S on the connection surface of three vertices A, B and C that are not on each side of a triangle or their extension lines intersect with edges BC, CA, AB or their extension lines at points P, Q and R, respectively, then BP/PC×CQ/QA×AR/RB= 1.

★28. The application theorem of Seva's theorem: Let the intersection of a straight line parallel to the side BC of △ABC and the two sides AB and AC BE D and E respectively, and let Be and CD intersect with S, then AS will pass through the center M of the side BC.

★29. Inverse theorem of Seva theorem: (omitted)

★30. Application Theorem of Inverse Theorem of Seva Theorem 1: The three midlines of a triangle intersect at one point.

★3 1, the inverse theorem of Seva Theorem Application Theorem 2: If the inscribed circle of △ABC is tangent to points R, S and T respectively, then AR, BS and CT intersect at one point.

★32. simonson's Theorem: Take any point P on the circumscribed circle of △ABC as the vertical line to three sides BC, CA, AB or their extension lines, and let its vertical feet be D, E and R respectively, then the line of D, E and R * * *, (this line is called simonson line).

★33. Inverse theorem of simonson's theorem: (omitted)

34. Steiner theorem: Let the vertical center of △ABC be H, which circumscribes any point p of the circle. At this time, the Simpson line of point p about △ABC passes through the center of the line segment pH.

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 about △ABC are: arc AP+ arc BQ+ arc Cr = multiple of 360.

37. The protagonist and Tengxia Theorem Inference 1: Let p, q and r be three points on the circumscribed circle of △ABC. If the Simonson lines of p, q and r about △ABC intersect at one point, then the Simonson lines of a, b and c about △PQR also intersect at the same point as before.

38. Inference 2 of Bolanger and Tengxia Theorem: In inference 1, the intersection of three Simpson lines is the midpoint of the line connecting the vertical centers of triangles made by any three points A, B, C, P, Q and R with the vertical centers of triangles made by other three points.

39. Bolanger and Tengxia Theorem Inference 3: Investigate the Simpson line of point P on the circumscribed circle of △ABC. If QR is perpendicular to this Simpson line, the Simpson lines about △ABC at P, Q and R will intersect at one point.

40. Inference 4 of Bolanger and Tengxia Theorem: Draw a vertical line from the vertex of △ABC to sides BC, CA and AB, let the vertical feet be D, E and F respectively, and let the midpoints of sides 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, and then L, M,

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 the three points of D, E and F are * * * lines.

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.

45. Qing Palace Theorem: Let P and Q be two points of the circumscribed circle of △ABC, which are different from A, B and C respectively. The symmetry points of point P about BC, CA and AB are U, V and W respectively. At this time, the intersection of the inflections, and the edges BC, CA, and AB or their extension lines are D, E, and F, respectively, then D, F

46. He takes the theorem: 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, if the intersections of flexion,, and edges BC, CA, AB or their extension lines are respectively ED, E and F, then D, E and F are three points. (anti-point: p and q are the radius OC of circle o and two points on its extension line, respectively. If OC2=OQ×OP, then the two points of P and Q are anti-points relative to the circle O)

47. Langerhans Theorem: There is a point A1B1C1D14 on the same circle. Take any three points as triangles, take a point P on the circumference, make a point P about Seymour lines of these four triangles, and then draw a vertical line from P to these four Seymour lines, then these four vertical feet are on the same straight line.

48. Draw a vertical line from the midpoint of each side of a triangle to the tangent of the circumscribed circle at the vertex of that side. These vertical lines intersect the center of the nine-point circle of the triangle.

49. There are n points on a circle, and the perpendicular lines drawn from the center of gravity of any n- 1 point to the tangents of other points of the circle intersect at one point.

50. Cantor Theorem 1: There are n points on a circle, and the perpendicular line drawn from the center of gravity of any n-2 points to the other two points is * * * points.

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 the Cantor line of M, N and L about pentagons A, B, C, D and E.

54. 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 and the midpoint of two diagonal lines, three * * * lines. This straight line is called Newton line of this quadrilateral.

57. Newton's Theorem 2: The midpoint, center and three-point * * * line of two diagonals of a circle circumscribed by a quadrilateral.

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 * * * lines.

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 * * * lines.

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 * * * points.

6 1, Basija theorem: A circle is inscribed in the intersection line (or extension line) of the opposite side AB of the hexagon ABCDEF and DE, BC and EF, CD and FA.