First, the triangle center of gravity theorem
Second, the external center theorem of triangle
Third, the triangle vertical center theorem
Fourthly, the interior theorem of triangle.
Proximity theorem of verb (verb abbreviation) triangle
A poem about the five hearts of a triangle
Five-center theorem of triangle
The center of gravity, outer center, hanging center, inner center and lateral center of a triangle are called the five centers of the triangle. The five-center theorem of triangle refers to the triangle's center of gravity theorem, outer center theorem, vertical center theorem, inner center theorem and near center theorem.
First, the triangle center of gravity theorem
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 at one point can be proved by the dovetail theorem, which is very simple. (The center of gravity was originally a physical concept. For a triangular thin plate with the same thickness and uniform mass, its center of gravity is just the intersection of the three midlines of the triangle, hence the name. The nature of the center of gravity is 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 center of gravity is equal to the area of the three triangles formed by any two vertices of the triangle. 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.
Second, the external center theorem of triangle
The center of the circumscribed circle of a triangle is called the outer center of the triangle. The property of the epicentre is 1, and the perpendicular lines of the three sides of the triangle intersect at a point, which is the epicentre of the triangle. 2. If O is the outer center of △ABC, ∠BOC=2∠A(∠A is acute angle or right angle) or ∠ BOC = 360-2 ∠ A (∠ A is 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 whose three vertices are connected with the other two vertices. c 1=d2d3，c2=d 1d3，C3 = d 1 D2； C=c 1+c2+c3. Eccentric 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.
Third, the triangle vertical center theorem
The three heights of a triangle intersect at a point, which is called the vertical center of the triangle. The nature of the vertical center: 1, three vertices and three vertical feet of a triangle. You can get six four-point circles by hanging these seven points. 2. Triangular three-point * * line of outer center O, center of gravity G and vertical center H, OG: GH = 1: 2. (This straight 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 proof shows that in Δ δABC, AD and BE are two heights, intersecting at point O, connecting CO and extending the intersection point AB to point F, proof: CF⊥AB proof: connecting de≈ADB =∠aeb = 90 degrees ∴A, B, D and E * * * circle ∠ Ade.
Fourthly, the interior theorem of triangle.
The center of the inscribed circle of a triangle is called the heart of the triangle. Intrinsic property: 1, the three bisectors of a triangle intersect at one point. This point is the center of the triangle. 2. The distance from the center to the right-angled triangle edge is equal to the sum of the two right-angled edges minus half the difference of the hypotenuse. 3.p is any point on the Δ ABC plane, and point 0 is the core of Δ ABC. The necessary and sufficient conditions are as follows: vector P0=(a× vector PA+b× vector PB+c× vector PC)/(a+b+c). 4.o is the core of a triangle, and A, B and C are the three vertices of the triangle respectively. When the intersection of AO and BC is extended to N, there are AO:ON=AB:BN=AC:CN=(AB+AC):BC 5, point O is any point on the plane ABC, and point I is △ABC. The necessary and sufficient conditions are: a (vector OA)+b (vector OB)+c (vector OC)= vector 0. 6. (euler theorem) ⊿. Then oi 2 = r 2-2rr.7, (the bisector of the inner corner is divided into three sides) △ABC, where 0 is the center, and the bisectors of the inner corners of ∠A, ∠B and ∠C intersect BC, AC and AB at Q, P, R and AB respectively.
Proximity theorem of verb (verb abbreviation) 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 edge center of the triangle. The nature of the edge center: 1, the bisector of the triangle-inner angle and the bisector of the outer angle at the other two vertices intersect at one 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 the centroid 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 side centers, and it 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.
A poem about the five hearts of a triangle
Triangle with five hearts (suspended and inside) A triangle has five hearts, suspended and inside. The essence of five minds is very important, so it is very important to master the confusion seriously. The three midlines of the center of gravity must intersect, and the intersection position is really strange. The intersection is named "center of gravity", and the nature of the center of gravity should be clear. The proportion of line segments in the division of center of gravity can be heard. The ratio of length to length is 2: 1, which can be used flexibly. An outer triangle has six elements, and three inner corners have three sides. Let three sides be perpendicular to each other and three lines intersect at a point * * *. This point is defined as the outer circle center and can be used as the circumscribed circle. Don't confuse internal and external centers, the key is to cut in and cut out. The three high points on the vertical center triangle must intersect with the vertical center. The high line is divided into three pairs of triangles and right angles. * * * The chart has four points, which can be found clearly after careful analysis. The inner triangle corresponds to three vertices, all angles and angles have bisectors, 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, which can be used as a triangle inscribed circle. The center of this circle is called "inner heart", so it is natural to define it. Don't forget the essence of five minds, so it's really good to start doing the problem.