1, the perpendicular bisector of each side of the triangle intersects at one point. 2. Pythagorean theorem 3. Three vertical lines drawn from each vertex of a triangle to its opposite side intersect at one point. 4. Projective Theorem (Euclid Theorem) 5. 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. 6. Let the outer center of the triangle ABC be O, the vertical center be H, draw a vertical line from O to BC, and let the vertical foot be M, then AH=2OM7, and the outer center, vertical center and center of gravity of the triangle are on the same straight line.

The extension of mathematical laws:

In the triangle 1, (nine-point circle or Euler circle or Fellbach circle), the center of three sides, the vertical foot drawn from each vertex to its opposite side, and the midpoint of the connecting line between the vertical foot center and each vertex are all on the same circle.

2. The straight line connecting the midpoints of two sides of the quadrilateral and the straight line connecting the midpoints of two diagonal lines intersect at one point.

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

4. euler theorem: The outer center, center of gravity, center of nine points and vertical center of a triangle are located on the same straight line (Euler line) in turn.

5. Coolidge's Maximum Theorem: (A nine-point circle inscribed with a quadrilateral)

There are four points on the circumference, any three of which are triangles, and the nine 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.

6. 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 = sqrt {[(s-a) (s-b)]/s} $ s is half the circumference of the triangle.

7. The bisector of the inner corner and the bisector of the outer corner of the triangle intersect at one point at the other two vertices.

8. Mean value theorem: (babs theorem) Let the midpoint of the side BC of triangle ABC be p, then there is $ AB 2+AC 2 = 2 (AP 2+BP 2) $.

9.Stewart Theorem: p If the side BC of the triangle ABC is divided into m:n, there is $ nxxab2+mxxac2 = (m+n) ap2+(Mn)/(m+n) bc2 $.

10, 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.

1 1, Apollonius theorem: the 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 of the line segment AB at both ends of the diameter.

12, Ptolemy theorem:

In the inscribed quadrilateral of a circle, the area of a rectangle surrounded by two diagonal lines is equal to the sum of the area of a rectangle surrounded by one set of opposite sides and the area of a rectangle surrounded by another set of opposite sides. From this theorem, the sum and difference formulas of sine and cosine and a series of trigonometric identities can be derived. Ptolemy theorem is essentially about the basic properties of circles.

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