Definition of "Four Hearts":
1, center of gravity: the intersection of three sides, and the center of gravity divides the length of the center line into 2: 1.
2. Vertical center: the intersection of three high lines, which are perpendicular to the corresponding edges.
3. Heart: The intersection of three bisectors (the center of the inscribed circle), and the distance from any point on the bisector to both sides of the corner is equal.
4. Outer center: the intersection of three vertical lines (the center of the circumscribed circle), and the distance from the outer center to each vertex of the triangle is equal.
Analyzing the Quad-centered Property by Mercedes-Ben Ci Theorem;
For △ABC, if O is a point on the △ABC plane, let O be a point on the △ABC plane, and the sides of inner angles A, B and C are A, B and C respectively, then:
1, center of gravity: If O is the center of gravity, according to similar knowledge, we can know that the areas of the three triangles divided by OA, OB and OC are the same, and they are all △ABC 1/3, then there is: vector OA+ vector OB+ vector OC= vector 0.
2. Outer center: If O is the outer center, we can know from the sine theorem of triangle that a/Sin∠A=b/Sin∠B=c/Sin∠C=2R, then OA = ob = oc = a/2sin ∠ a = b/2sin ∠.
3. Inner heart: If O is the inner heart and the circle is tangent to the three sides of △ABC, then the area of three small triangles can be expressed by multiplying the bottom by the height, and the same height is the radius of the circle, then the area ratio of three small triangles is equivalent to the ratio of the bottom, that is, S △ BOC: S △ AOC: S △ OA+b = A: B: C. According to Mercedes-Ben Ci theorem, a conclusion can be drawn.
4. Vertical center: if O is the vertical center, vector OA tan∠A+ vector OB tan∠B+ vector OC tan ∠ C = 0.