How to judge the center of triangle

Triangles have five centers: inner center: the intersection of the three bisectors, which is also the center of the inscribed circle of the triangle. Properties: the distance to the three sides is equal. Outer center: the intersection of the three perpendicular lines, which is also the center of the circumscribed circle of the triangle

What do triangle center, center of gravity, outer center, inner center and perpendicular center mean respectively?
By the way, what's the other side

Center of gravity: the intersection of the center line perpendicular: the intersection of the high (vertical line) outer center: the center of the circumscribed circle of the triangle, that is, the intersection of the vertical bisector of the edge inner center: the intersection center of the inscribed circle of the triangle, that is, the intersection center of the angular bisector: that is, the geometric center, which is mainly mentioned in the centrosymmetric figure: the outer angle of any two corners of the triangle

What are the properties of the center of a triangle?

The center is also the center of gravity of the midpoint triangle;

1. It is known that the vertex coordinates of the image of quadratic function y = ax, square + BX + C are (- 1, - 4), the abscissa of its two intersections with the X axis are X1 and X2 respectively, and the square of X1 + the square of x2 = 10
2. It is known that the image of quadratic function y = ax square + BX + C and the image of function y = - 2x square + 4x + 1 are symmetric about X axis. The analytic expression of the quadratic function is obtained
3. It is known that the symmetry axis of the square + BX + C of the parabola y = x is on the right side of the y-axis, and the intersection of the parabola and the y-axis is Q (0, - 3), the intersection of the parabola and the x-axis is a, B, the vertex is p, and the area of the triangle APB is 8

1 B / 2A = - 14ac-b square / 4A = - 4x1 square + x2 square = (x1 + x2) 2-2x1x2 = (- B / a) 2-2 * C / a = 10 three equations three unknowns 2 about X axis symmetry coefficient corresponding to a = - 2 b = - 4 C = - 13 symmetry axis on the right side of Y axis, we can know - B / 2a is greater than 0, a = 1, so B is greater than 0, the intersection of object line and Y axis is Q (0, - 3) we can know C = - 3P

Quite a simple high school quadratic function inequality problem, can't solve it!
X belongs to the square of R, y = x, and the value of - 4ax + 2a-30 is not negative

Y = x & sup2; - 4ax + 2a-30 is nonnegative, which means that its minimum value is ≥ 0
y=x²-4ax+2a-30
=(x-2a)²-4a²+2a-30
So the minimum value is - 4A & sup2; + 2a-30 ≥ 0
That is 2A & sup2; - A + 15 ≤ 0
There is no solution
There must be something wrong with the title. Please check it yourself

A function problem needs to be solved
Given that the length of the hypotenuse of a right triangle is 16cm, then the length of a right triangle is The area of the right triangle is the largest, and the maximum area is_ .
Need process

Area s ^ 2 = 1 / 4 times x ^ 2 times (16 ^ 2-x ^ 2), let x ^ 2 = y. then s ^ 2 = 1 / 4 times y times (16 ^ 2-y), you can get the value of y when the square of the area is the largest, that is, the value of y when the area is the largest, so you can get the value of X is 8 times the root 2, and the area is 64

The process of solving a mathematical function problem
Straight line y = 3x + 6, respectively, through the first, second and third quadrants, there is a point a (- 6,0) on its right side, and there is a moving point P on this straight line. When p (x, y) moves to what position, the area of triangle OPA = 27 divided by 8
Or someone who doesn't understand can directly help me find out (3x / 4 + 6) times x = 27 / 16, find X

You just mark point a, mark m (0,6) on the y-axis, mark n (- 8,0) on the x-axis, and connect Mn. Mn is a straight line with y = 3x / 4 + 6
The area calculation of triangle OPA is very simple. Do you notice that the bottom is not changed, OA = 6, directly multiplied by Y * 0.5 equals 27 / 8
That is, 3Y = 27 / 8, y = 9 / 8
Back to the equation, x = - 13 / 2
I don't know how you can work out the equation (3x / 4 + 6) * x = 27 / 16?

The process of solving several high school function problems
1. If f (x) = x & sup2; + (3a-2) x + (A-1) has a zero point on [1,3] and only one zero point on the x-axis, then the range of real number a is small
2. Let f (x) = 2x & sup3; + BX + C (b > 0), (- 1 ≤ x ≥ 1), f (- 1 / 2) f (1 / 2)

1. To ensure that there is a zero on [1,3] and x-axis, only f (1) * f (3) is less than or equal to 0, and the value of a whose f (1) and f (3) are equal to 0 at the same time should be deleted. 2. The derivation of function f (x) = 2x & sup3; + BX + C (b > 0) shows that its derivative function is in the interval [- 1,1] and is greater than 0

A company went to the orchard base to buy some high-quality fruit to express sympathy to the medical workers. The orchard base has two sales plans for the customers who purchase more than 3000 kg (including 3000 kg): 9 yuan per kilogram, and the base will deliver it to the door; scheme B: 8 yuan per kilogram, and the customers will rent a car to transport it back. It is known that the transportation cost of the company's rental car from the base to the company is 5000 yuan The functional relationship between the payment y yuan of the two purchase schemes and the quantity x kg of fruit purchased by the company is given, and the value range of the independent variable x is written out; (2) when the purchase quantity is in what range, the payment of scheme a is the least; when the purchase quantity is in what range, the payment of scheme B is the least, and the reasons are explained

(1) Scheme A: Y1 = 9x (x ≥ 3000)), scheme B: y2 = 8x + 5000 (x ≥ 3000)); & nbsp; & nbsp; (2) ① when Y1 < Y2; 9x < 8x + 5000, x < 5000; ② when Y1 > Y2; 9x > 8x + 5000, x > 5000; answer: when the fruit weight is less than 5000 kg and more than 3000 kg

How to judge the center of triangle