Triangle center of gravity theorem
\x05 The median lines of three sides of a triangle intersect at one point. This point is called the center of gravity of the triangle.
The intersection of three median lines can be proved by dovetail theorem, which is very simple. (The center of gravity was originally a physical concept. For a triangle slice with the same thickness and uniform quality, its center of gravity is exactly the intersection of the three midlines of this triangle, hence the name. )
Nature of the center of gravity:
1, and the ratio of the distance from the center of gravity to the vertex to the distance from the center of gravity to the midpoint of the opposite side is 2: 1.
2. The area of the three triangles composed of the center of gravity and the three vertices of the triangle is equal, that is, the distance from the center of gravity to the three sides is inversely proportional to the growth of the three sides.
3. The sum of squares of the distances from the center of gravity to the three vertices of the triangle is the smallest.
4. In the plane rectangular coordinate system, the coordinate of the center of gravity is the arithmetic average of the vertex coordinates, that is, the coordinate of the center of gravity is ((x 1+x2+x3)/3, (y 1+y2+y3)/3.
\x05 II。 Triangle eccentricity theorem
\x05 The center of the circumscribed circle of a triangle is called the outer center of the triangle.
The nature of the external world:
1, the perpendicular lines of the three sides of the triangle intersect at a point, which is the outer center of the triangle.
2. If O is the outer center of △ABC, ∠BOC=2∠A(∠A is an acute angle or a right angle) or ∠ BOC = 360-2 ∠ A (∠ A is an obtuse angle).
3. When the triangle is an acute triangle, the outer center is inside the triangle; When the triangle is an obtuse triangle, the outer center is outside the triangle; When the triangle is a right triangle, the outer center is on the hypotenuse and coincides with the midpoint of the hypotenuse.
4. To calculate the coordinates of the epicenter, we must first calculate the following temporary variables: D 1, D2 and D3 are the point multiplication of the vectors connecting the three vertices and the other two vertices of a triangle respectively. C 1 = D2D3，C2 = D 1D3，C3 = d 1 D2； C=c 1+c2+c3。 Gravity center coordinates: ((C2+C3)/2c, (C 1+C3)/2c, (C 1+C2)/2c).
5. The distances from the outer center to the three vertices are equal.
\x05 three. Vertex theorem of triangle
\x05 The three heights of a triangle intersect at a point, which is called the vertical center of the triangle.
The essence of the heart:
1, a triangle with three vertices and three vertical feet can get six four-point circles.
2. The three-point * * line of the triangle outer center O, center of gravity G and vertical center H, OG∶GH= 1∶2. This line is called the Euler line of triangle. )
3. The distance from the vertical center to the vertex of the triangle is twice as long as the distance from the outer center of the triangle to the opposite side of the vertex.
The product of two parts of each high line is equal.
theorem proving
It is known that in Δ δABC, AD and BE are two heights, which intersect at point O, connect CO and extend the intersection of AB at point F. Verification: CF⊥AB.
Prove:
Connect de≈ADB =∠aeb = 90 degrees ∴A, B, D and E * * * cycles ∴∠ADE=∠ABE.
∠∠eao =∠DAC∠AEO =∠ADC ∴δaeo∽δadc
∴ae/ao=ad/ac ∴δead∽δoac ∴∠acf=∠ade=∠abe
∠∠Abe+∠BAC = 90 degrees ∴∠ACF+∠BAC=90 degrees ∴∴ CF ⊥ AB.
Therefore, the vertical center theorem holds!
\x05 four. Interior theorem of triangle
The center of the inscribed circle of a triangle is called the heart of the triangle.
Intrinsic essence:
1, the three bisectors of the triangle intersect at one point, which is the heart of the triangle.
2. The distance from the center to the edge of a right triangle is equal to the sum of two right angles minus the difference of the hypotenuse.
3.p is any point on the plane of Δ ABC, and the necessary and sufficient conditions for point I to be the heart of Δ ABC are: vector PI=(a× vector PA+b× vector PB+c× vector PC)/(a+b+c).
4.o is the heart of the triangle, and A, B and C are the three vertices of the triangle. If the edge of the AO intersection BC extends to n, there is AO:ON=AB:BN=AC:CN=(AB+AC):BC.
\x05 V. Proximity Theorem of Triangle
The center of the tangent circle of a triangle (the circle tangent to the extension line of one side and the other two sides of the triangle) is called the tangent center of the triangle.
Nature of lateral center:
1, the bisector of an inner corner of a triangle and the bisector of an outer corner of the other two vertices intersect at a point, which is the edge center of the triangle.
2. Every triangle has three side centers.
3. The distance from the side center to the three sides is equal.
As shown in the figure, point M is an apocentric point of △ABC. The intersection of the bisector of the outer angle of any two angles of a triangle and the bisector of the inner angle of the third angle. A triangle has three centers and must be outside the triangle.
Attachment: the center of a triangle: only a regular triangle has a center. At this time, the center of gravity, inner heart, outer heart, hanging heart and four hearts are integrated.
\x05 poem about five hearts of a triangle; The Song of the Five Hearts of the Triangle (with emphasis on the outside and inside)
There are five hearts in a triangle, and five hearts are very important.
Chongxin
The three midlines must intersect, and the location of the intersection is really strange. The intersection is named "center of gravity", and the nature of the center of gravity should be clear.
In the center of gravity segmentation, you can hear the ratio of line segment to several segments; The ratio of length to length is two to one, so we should use it flexibly and master it well.
External heart
A triangle has six elements, three inner corners and three sides. Let three sides be perpendicular to each other and three lines intersect at one point.
This point is defined as the epicentre and can be used as the circumscribed circle.
worry (about/over)
If a triangle is three high points, the three high points must intersect at the vertical center. The high line is divided into triangles with three pairs of right angles.
There are twelve right triangles, forming six pairs of similar shapes, which can be found in the four-point * * * diagram. Careful analysis can clearly find them.
heart
A triangle corresponds to three vertices, each corner has a bisector, and the three lines intersect at a certain point, which is called "inner heart" and has roots;
The points to the three sides are equidistant and can be inscribed into a triangle. This center of the circle is called "inner heart", so it is natural to define it.