The intersection of the three midlines of a triangle is called the center of gravity of the triangle.

Theorem: Let the center of gravity of triangle be O and the midpoint of BC side be D, then there is AO = 2 OD.

The coordinates of the center of gravity are the average of the coordinates of three vertices.

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2. Outside the heart

The intersection of the perpendicular lines of the three sides of a triangle is called the outer center of the triangle.

The distances from the outer center to the three vertices are equal.

The circle passing through the vertex of a triangle is called the circumscribed circle of the triangle, and the center of the circumscribed circle is the outer center of the triangle. This triangle is called the inscribed triangle of this circle.

A triangle has one and only one circumscribed circle.

Eccentricity formula:

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3. Heart

The center of a triangle is the intersection of bisectors of three internal angles of the triangle.

A circle tangent to all sides of a triangle is called the inscribed circle of the triangle. The center of the inscribed circle is the center of the triangle, and the distances from the center to the three sides of the triangle are equal. This triangle is called the circumscribed triangle of a circle.

A triangle has one and only one inscribed circle.

Internal coordinate formula:

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4. Hold on to your heart

The three high lines of three sides of a triangle intersect at a point, which is called the vertical center of the triangle.

The vertical center of an acute triangle is within the triangle; The vertical center of a right triangle is at the vertex of the right angle; The vertical center of an obtuse triangle is outside the triangle.

A triangle has only one center.

Ordinate formula:

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5. Lateral center

The circle tangent to the extension line of one side and the other two sides of a triangle is called the tangent circle of the triangle, and the center of the tangent circle is called the edge center of the triangle.

A bisector of the inner corner of a triangle intersects with the bisectors of the outer corners of the other two corners at a point, that is, the center of the triangle. The distance between the extension lines of one side and the other two sides of a triangle and the center of the side is equal.

A triangle has three tangent circles and three edge centers. The distances between the centers of the three sides of a triangle and the extension lines of the three sides are equal.

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The essence of five minds

The five centers of a triangle have many important properties, which are also closely related, such as:

(1) The center of gravity of the triangle is equal to the area of the three triangles formed by the connecting lines of the three vertices;

(2) The distances from the outer center of the triangle to the three vertices are equal;

(3) Of the four points and three vertices of the triangle's vertical center, any point is the vertical center of the triangle formed by the other three points;

(4) The distances from the centers of the inner and outer sides of the triangle to the three sides are equal;

(5) The vertical center of a triangle is the center of its vertical triangle; In other words, the heart of a triangle is the center of the triangle next to it;

(6) The outer center of the triangle is the vertical center of the midpoint triangle;

(7) The center of gravity of the triangle is also the center of gravity of the midpoint triangle;

(8) The center of a triangle is also the center of its vertical triangle.

(9) The distance from any vertex of a triangle to the vertical center is equal to twice the distance from the outer center to the opposite side.

The following are more detailed properties:

1, relax

The intersection of the heights of three sides of a triangle is called the vertical center of the triangle. The vertical center of a triangle has the following interesting properties: let the three heights of △ABC BE AD, be and CF, where d, e and f are vertical feet and the vertical center is H.

The symmetry points of real estate 1 vertical center h on three sides are all on the circumscribed circle of △ABC.

In the property 2 △ABC, there are six groups of four-point * * * circles and three groups of similar right-angled triangles, ah HD = BH He = CH HF.

Any one of the four points of properties 3 H, a, b and c is the vertical center of a triangle with the other three points as its vertices (such four points are called vertical center groups).

The circumscribed circle of properties 4 △ ABC, △ abh, △ BCH and △ ach is an isometric circle.

Property 5 In a non-right triangle, if the straight line passing through H intersects AB and AC in P and Q respectively, AB/AP Tanb+AC/AQ Tanc = Tana+Tanb+Tanc.

The distance from any vertex of a triangle to the vertical center is equal to twice the distance from the outer center to the opposite side.

Property 7 Let O and H be the external center and vertical center of △ABC respectively, then ∠BAO=∠HAC, ∠ABH=∠OBC, ∠BCO=∠HCA.

Property 8 The sum of the distances from the vertical center to the three vertices of an acute triangle is equal to twice the sum of the radii of its inscribed circle and circumscribed circle.

Property 9 The vertical center of an acute triangle is the center of a vertical triangle; Among the inscribed triangles of acute triangle (the vertex is on the side of the original triangle), the perimeter of vertical triangle is the shortest.

2. Heart

The center of the inscribed circle of a triangle is simply called the center of the triangle, that is, the intersection of bisectors of three angles of the triangle. The heart has the following beautiful characteristics:

Property 1 Let I be the heart of △ABC, then I is the heart if and only if the distances to the three sides of △ABC are equal.

Property 2 Let I be the heart of △ABC, then ∠ BIC = 90+ 1/2 ∠ A, and there are two similar expressions; or vice versa, Dallas to the auditorium

Property 3 Let I be a point in △ABC, the line where AI is located intersects the circumscribed circle of △ABC at point D, and the necessary and sufficient condition for I to be △ABC is ID=DB=DC.

Property 4 Let I be the heart of △ABC, BC=a, AC=b, AB=c, and the projections of I on BC, AC and AB are D, E and F respectively; If the radius of the inscribed circle is r, p= (1/2)(a+b+c), then (1) s △ ABC = pr; (2)r = 2S△ABC/a+b+ c； (3)AE=AF=p-a，BD=BF=p-b，CE = CD = p-c； (4)abcr=p Abici.

Property 5 The distance from the intersection of the bisector of a triangle and its circumscribed circle to the other two vertices is equal to the distance from the center; On the other hand, if I is the point on the bisector AD of △ABC (D is on the circumscribed circle of △ABC) and DI=DB, then I is the heart of △ABC.

Property 6 Let I be the heart of △ABC, BC=a, AC=b, AB=c, ∠A intersects BC at K, and the circumscribed circle of △ABC intersects D, then AI/Ki = AD/DI = DI/DK = (B+C)/A.

3. Outside the heart

The center of the circumscribed circle of a triangle is simply called the outer center of the triangle, that is, the intersection of the three vertical lines of the triangle. The external center has the following series of beautiful features:

The distance from the outer center of the attribute 1 triangle to the three vertices is equal, and vice versa.

Property 2 Let O be the outer center of △ABC, then ∠BOC=2∠A, or ∠ BOC = 360-2 ∠ A (there are two other formulas).

Property 3 Let three sides of a triangle be long, and the radius and area of the circumscribed circle are A, B, C, R and S△ respectively, then R=abc/4S△.

Property 4 If the epicenter o of △ABC passes through a straight line and the sides of AB and AC (or extension line) intersect at P and Q points respectively, then AB/AP SIN2B+AC/AQSIN2C = SIN2A+SIN2B+SIN2C.

Property 5 The sum of the distances from the outer center of an acute triangle to three sides is equal to the sum of the radii of its inscribed circle and circumscribed circle.

4. Center of gravity

Property 1 Let G be the center of gravity of △ABC, and the projection of point Q on BC, CA and AB in △ABC is D, E and F, respectively, then QD QE QF is the largest when Q and G coincide; or vice versa, Dallas to the auditorium

Property 2 Let G be the center of gravity of △ABC, and the extension lines of AG, BG and CG intersect with the three sides of △ABC at D, E and F, then S △ AGF = S △ BGD = S △ CGE; or vice versa, Dallas to the auditorium

Property 3 Let G be the center of gravity of △ABC, then S △ ABG = S △ BCG = S △ ACG = (1/3) S △ ABC; or vice versa, Dallas to the auditorium

5. 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.