The vertical center, center of gravity and outer center of the same triangle are three-point * * * lines, which are called Euler lines of the triangle; And the distance between the outer center and the center of gravity is equal to half the distance between the vertical center and the center of gravity.

2. Nine o'clock circle:

The midpoint of three sides of an arbitrary triangle, the three-high vertical foot and the midpoint of the line segment between the three vertices and the vertical center, and the * * * nine-point * * circle are called the nine-point circle of the triangle; Its center is the midpoint of the connecting line between the outer center of the triangle and the vertical center, and its radius is equal to half the radius of the circumscribed circle of the triangle.

3. fermat point:

It is known that p is a point within the acute angle △ABC. When ∠ APB = ∠ BPC = ∠ CPA = 120, the value of PA+PB+PC is the smallest, and this point P is called the fermat point of △ABC.

4.Heron formula:

△ABC, the lengths of BC, CA and AB are A, B and C respectively. If p = (A+B+C), the area of △ABC is S.

5.Ceva theorem:

In △ABC, if the vertex crossing △ABC is a straight line intersecting point P, and the sides BC, CA and AB intersect points D, E and F respectively, then; Its reverse is also correct.

6. Meeker Point:

If four straight lines AE, AF, ED and FB intersect at six points A, B, C, D, E and F to form four triangles, namely △ABF, △AED, △BCE and △DCF, then the circumscribed circle of these four triangles is the * * * point, which is called Miguel point.

7. Gergonne Point:

The inscribed circle of △ABC is at point D, point E, point F, point AB, point BC and point CA respectively, so point AE, point BF and point CD are triple * * * points, which are called Guegan points.

8. siemsen Line:

It is known that P is any point on the circumscribed circle of △ABC, and PD⊥BC, PE⊥ACPF⊥AB, D, E and F are vertical feet, then the three points of D, E and F are * * * lines, which are called Simpson lines.

9, the golden section:

A line segment (AB) is divided into two lines so that the larger line segment (AC) is the median of the ratio of the original line segment (AB) to the smaller line segment (BC). This division is called the golden section.

10, Pythagorean theorem:

That is to say, the sum of squares of two right angles of a right triangle is equal to the square of the hypotenuse. This is the most basic and important theorem in plane geometry, which is called Pythagoras theorem abroad.

1 1, Gillard Girard Desargues theorem:

It is known that in △ ABC and △A'B'C, three lines AA', BB' and CC' intersect at points O, BC and B' c', CA and C'A', AB and A'B' intersect at points X, Y and Z respectively, so three points X, Y and Z are * * * lines; Its inverse is also true.

12, Molly Triangle: In the bisector of the known triangle △ABC, every two straight lines adjacent to BC, CA and AB intersect at points D, E and F, then the triangle DDE is a regular triangle, which is called Molly Triangle.

13, Pascal's theorem: It is known that the extension lines of AB side and DE side inscribed with hexagonal ABCdeF intersect at G point, BC side and EF side intersect at H point, and CD side and FA side intersect at K point, then the three points of H, G and K are * * * lines.

14, Ptolemy theorem:

In a quadrilateral inscribed in a circle, AB? CD+AD？ BC=AC？ Bachelor of science

15, the ratio of the distance from the fixed point P to the two fixed points A and B in the Apollonius circle is equal to the fixed ratio m:

N, then the locus of point P is a constant ratio m:

The straight line connecting the N bisector and the two bisectors of the N bisector is a circle with a diameter, which is called the Apollonius circle, or Arrhenius circle for short.

16, Menelaus theorem

17, Brahma Gupta theorem:

In a quadrilateral ABCD inscribed in a circle, AC⊥BD is perpendicular to one side from the diagonal intersection P, and its extension line will bisect the opposite side.