Mathematical geometric theorem and 13 basic inequalities

Well-known theorem in ry (not sure all are correct, but some errors have been found and corrected)
1. Pythagorean Theorem)
2. Projection Theorem (Euclidean theorem)
3. The three midline of the triangle is handed over to a point, and each midline is divided into two parts by this point: 2: 1.
4. The link between the two diagonal centers of the Center on both sides of the quadrilateral is placed at one point.
5. The center of the two triangles made by the center of the side connecting the hexagonal edges are overlapped.
6. The vertical and one alignment of each side of the triangle is handed over to a point.
7. Place the three vertical lines from the vertex of the triangle to the edges at a point.
8. If the outer center of the Triangle ABC is O, the center is H, and the vertical line is directed from O to the BC side. If the perpendicular foot is not L, AH = 2OL
9. The outer heart of the triangle, with the center of gravity in the same line.
10. In a triangle (9-dot circle, Euler's circle, or ferbach circle), the center of the three sides, the vertical foot of the vertical line directed to the opposite side from each vertex, And the midpoint of the link between the center and each vertex, these nine points are on the same circle,
11. Euler's theorem: the outer heart, center of gravity, center of the nine-point circle, and center of the triangle are in the same line (Euler's line) in sequence.
12. Ku LiQi * big upper theorem: (9-Point Circle of the inner quadrilateral)
There are four points in the circumference, and any three points in the circle are used as triangles. The nine-point circle of these four triangles is located in the same circumference. We call the circle of the four nine-point circle the circle of the circle the nine-point circle of the circle.
13. The three inner-angle bisector of the (inner) triangle is handed over to a point. The Radius Formula of the Inner-angle triangle is r = (s-a) (s-B) (s-c) ss is half of the triangle perimeter
14. An inner-angle bisector of the triangle and the outer-angle bisector of the other two vertices are handed in at one point.
15. midline theorem: (buss theorem) if the midpoint of the edge BC of the Triangle ABC is P, then AB2 + AC2 = 2 (AP2 + BP2)
16. Stuart's theorem: P divides the edge BC of the Triangle ABC into m: n, and n x AB2 + m x AC2 = (m + n) AP2 + mnm + nBC2
17. Boro Moo and multi-theorem: When the diagonal lines of the Quadrilateral ABCD are perpendicular to each other in the circle, the straight lines connecting the M point of the AB and the E point of the diagonal line are perpendicular to the CD
18. Apollo NEIS theorem: the ratio of the distance to A and B at the two fixed points is the point P of the fixed ratio m: n (the value is not 1), which is located in dividing the line AB into m: the inner point C and the outer point D of n are the fixed circumference of the two vertices in the diameter.
19. torle's theorem: If a quadrilateral ABCD is connected to a circle, AB x CD + AD x BC = AC x BD
20. Take the edge BC, CA, and AB of any Triangle ABC as the bottom edge, and make the bottom angle of the triangle respectively 30 degrees equal waist △bdc, △cea, △afb, then △def is a positive triangle,
21. Love's Theorem 1: If △abc and delta are both normal triangles, the triangle formed by the center of gravity of the line segments AD, BE, and CF is also a positive triangle.
22. Love's Theorem 2: If △abc, △def, and △ghi are all positive triangles, then, the triangle that consists of the center of gravity of the triangle △adg, △beh, and △cfi is a positive triangle.
23. menelos theorem: Set the three sides of △abc, BC, CA, AB, or its extension line, and the intersection of a straight line without passing through any of their vertices to P, Q, and R.
BP/PC × CQ/QA × AR/RB = 1
24. Inverse Theorem of the menegius theorem: (omitted)
25. Application Theorem 1 of the menegus theorem: Let's set the cross edge of the outer angle bisector of △abc's limit A to the split edge of Q and limit C to R, if the cross edge of the bisector B is CA in Q, P, Q, and R are collocated.
26. Application Theorem 2 of the menegarus theorem: returns the tangent of the outer circular of any third vertex A, B, and C of △abc, and BC, CA, and AB extension line at the point P, Q, R, P, Q, and R are collocated.
27. Seva's theorem: Set the three vertices A, B, and C of △abc to three straight lines formed by A point S connecting surface on the side of the triangle or their extended line, BPPC × CQQA × ARRB () = 1.
28. Application theorem of the Seva theorem: Let the intersection of the straight lines parallel to the edge BC of △abc and the AB and AC on both sides BE D and E, and let BE and CD BE handed over to S, then, AS must be the center M of the Cross-edge BC
29. Inverse Theorem of the Seva theorem: (omitted)
30. Application of the Inverse Theorem of the Seva Theorem 1: The three midline of a triangle is handed over to a point
31. Application Theorem 2 of the Inverse Theorem of the Seva theorem: Let the incircle and edge BC, CA, and AB of △abc be tangent to points R, S, and T, respectively, then, AR, BS, and CT are handed over to one point.
32. westmoron's theorem: Starting from the outer circle of △abc, P laid a vertical line toward the third-side BC, CA, AB, or its extended line, and set its perpendicular feet to D, E, and R, respectively, then D, E, and R are all in the same line (this line is called the West mosong line)
33. Inverse Theorem of the westmoron theorem: (omitted)
34. Stana's theorem: Let's set the center of △abc to H, and set it to any point P of its outer circle. At this time, the westmoron line of P of △abc passes through the center of the PH of the line segment.
35. Application theorem of the Stanner theorem: on the outer circle of △abc, the symmetric point of P on the edge BC, CA, and AB is the same as the center H of △abc (parallel to the westmoron line. This line is called the image line of point P about △abc.
36. polangjie and tengxia theorem: Let the three points on the outer circle of △abc be P, Q, R, the necessary and sufficient conditions for P, Q, and R to place △abc at a point are: arc AP + arc BQ + arc CR = 0 (mod2 regression ).
37. Wave langjie and tengxia theorem inference 1: Set P, Q, and R to three points on the outer circle of △abc, if P, Q, and R are handed over to one point on the west Moson line of △abc, then the West Moson lines of A, B, and C about △pqr are handed over to the same point as before.
38. boranjie and tengxia theorem inference 2: In inference 1, the intersection of the three westmoron pine lines is the midpoint of the Center of A, B, C, P, Q, and R at any of the three points and the center of the other three points.
39. Wave langjie and tengxia theorem inference 3: Examine the westmoron line of P on the outer circle of △abc, for example, if the QR code is set to the string perpendicular to the external ballpoint pen of the westmoron pine line, the three-point west moron pine lines of P, Q, and R are handed over to one point.
40. Wave langjie and tengxia theorem inference 4: Starting from the vertex of △abc to the vertical line of the edge BC, CA, and AB, set the perpendicular foot to D, E, F, respectively, if the center of edge BC, CA, and AB are L, M, and N respectively, then D, E, F, L, M, and N are in the same circle, at this time, the westmoron line of △abc is handed over to one point.
41. Theorem 1 of the westmoron line: the two endpoints P and Q of the outer circle of △abc are perpendicular to each other, and their intersections are on the nine-point circle.
42. Theorem 2 (Anning theorem) of the westmoron pine line: There are four points in a circle, with any three points as triangles, make the remaining thymo pine lines about the triangle, and these thymo pine lines are handed over to one point.
43. KANO theorem: A point P of the outer circle of △abc is used to guide the three sides of △abc, namely, BC, CA, and AB, to form a straight line with the same direction, such as PD, PE, and PF, if the intersection of the three sides is D, E, and F, the three points of D, E, and F are collocated.
44. Oracle theorem: through the three vertices of △abc, we can draw three parallel straight lines and set the intersection points of them and the outer circle of △abc to L, M, N, respectively, if a point of P is obtained from the outer circle of △abc, the intersection points of PL, PM, PN, and △abc are D, E, and F, respectively, then D, E, and F are collocated.
45. Qing Gong's theorem: Let P and Q be the outer circular of △abc, which is different from the two points of A, B, and C, the symmetry points of point P on the three sides of BC, CA, and AB are U, V, and W, the intersection of QU, QV, QW, and edge BC, CA, and AB or their extended cables are D, E, and F, and the three points of D, E, and F are collocated.
46. His theorem: Let P and Q be a pair of inverse points about the outer circle of △abc, the symmetry points of point P on the three sides of BC, CA, and AB are U, V, and W, if the intersection of QU, QV, and QW with edge BC, CA, AB, or its extension line is ED, E, and F, then the three points of D, E, and F are all in line. (Inverse point: P and Q are the radius OC of the circle O and the two points of its extension line respectively. If OC2 = OQ × OP, P and Q are called the inverse points of the circle O)
47. longolai's theorem: There are A1B1C1D14 points on the same circle, where three points are used as triangles, and a point P is taken on the circumference. Then, the four points are used as the westmoron line of the four triangles, then, draw a vertical line from P to the four westmoron lines, and the four vertical feet are on the same line.
48. Draw a vertical line from the midpoint of each side of the triangle to the tangent of the outer circle at the vertex of the edge. These vertical lines are handed over to the center of the nine-point circle of the triangle.
49. There are n points on a circumference. from the center of gravity of any n-1 points in the circumference, the vertical lines cited by the tangent at the other points of the circumference are all handed over to one point.
50. Conway's Theorem 1: There are n points in a circle, and the center of gravity of any point in the n-2 is directed to the common point of the vertical line of the remaining two points.
51. Conway's Theorem 2: There are four points in A circumference: A, B, C, D, and M and N, then the M and N points of the four triangles △bcd, △cda, △dab, △abc in each of the two westmoron Pine Point on the same line. This line is called the "M" and "N" points on the "X-ray ABCD" line.
52. Conway's Theorem 3: There are four points on A circumference, B, C, D, and three points on M, N, and L, then the X-ray, L, and N-points of the X-ray ABCD and the X-ray, M, and L-points of the X-ray ABCD are handed over to one point. This point is called M, N, L, and so on.
53. Conway's theorem 4: There are five points on A circumference, B, C, D, E, and three points on M, N, and L, then, the M, N, and L points on each of the Quadrilateral BCDE, CDEA, DEAB, and EABC points in a straight line. This line is called M, N, L three points on the Pentagon A, B, C, D, E of the Conway line.
54. ferbach's theorem: the nine-point circle of a triangle is tangent to the incircle and the subtangent circle.
55. Mo Li's theorem: when the three inner and third-point points of a triangle are close to the two third-point points of an edge to obtain an intersection point, such three intersections can form a positive triangle. This triangle is often called the merley triangle.
56. Newton's Theorem 1: the point of the intersection of the two sides of the Quadrilateral edge, the center of the connected segment and the midpoint of the two diagonal lines, and the three colons. This line is called the Newton line of this quadrilateral.
57. Newton's Theorem 2: the midpoint of the two diagonal lines of the outer tangent Quadrilateral of the circle, the center of the circle, and the three-point collinearity.
58. dishag Theorem 1: There are two triangles △abc and △def on the plane, and their corresponding vertices (A and D, B, E, C, and F) if the corresponding edge or its extension line is intersecting, the three intersections are collocated.
59. Disha lattice Theorem 2: two triangles △abc and △def exist on the different planes, and their corresponding vertices (A and D, B, E, C, and F) are set) if the corresponding edge or its extension line is intersecting, the three intersections are collocated.
60. bryansong's theorem: links the vertex A, D, B, E, C, and F relative to the hexagonal ABCDEF of the circle, then the three-line common point.
60. Barrier theorem: the intersection of the edge AB and DE, BC, EF, CD, and FA of the hexagonal ABCDEF in the circle is the same

13 basic inequalities