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 side lengths BC, CA and AB are A, B and C respectively. If p = (A+B+C),

Then 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, the sum of the 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:

If every two adjacent lines of BC, CA and AB intersect at points D, E and F in the bisector of known internal angle △ABC, the triangle DDE is a regular triangle, which is called Mali triangle.

13, Pascal theorem:

It is known that the extension lines of AB and DE inscribed with hexagonal ABCdeF intersect at G point, BC and EF intersect at H point, and CD and FA 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? 6? 1CD+AD？ 6? 1BC=AC？ 6? 1BD

Apollonius Square 15

When the ratio of the distance between the moving point P and the two fixed points A and B is equal to the fixed ratio M: N, the trajectory of the point P is a circle with the diameter of the connecting line between two points of the fixed line segment with the fixed ratio M: N, 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.