What are the center, the perpendicular, the center and the center of gravity of the triangle?

What are the center, the perpendicular, the center and the center of gravity of the triangle?

The distance from this point to the vertex is twice the distance from it to the midpoint of the opposite side. (theorem of center of gravity). This intersection is called the center of gravity of the triangle. The vertical bisectors of the three sides of the triangle intersect at a point. (theorem of outer center) this point is called the outer center of the triangle

What are the center, center of gravity, center of perpendicularity, outer center and inner center of a triangle?
What other "hearts" are there?

The center of gravity theorem of IANA triangle: the three middle lines of a triangle intersect at a point, and the distance from this point to the vertex is twice the distance from it to the midpoint of the opposite side. This point is called the center of gravity of a triangle. The outer center theorem: the vertical bisectors of the three sides of a triangle intersect at a point

How to distinguish the center center of a triangle from the center of gravity
How to draw

Center of gravity: the intersection of the three midlines
Perpendicular: the intersection of three high lines
Outer center: the intersection of the vertical bisectors of three sides
Heart: the intersection of three bisectors
If and only if a triangle is an equilateral triangle, the center of an equilateral triangle is the four centers

(1) Given that f (x) = 2 ^ x, G (x) is a linear function, Let f (x) = f [g (x)], and the point (2,1 / 4) is on both the image of F (x) and the image of f ^ - 1 (x), then the analytic expression of F (x) is?
(2) Given the function f (x) = (a ^ x-1) / (a ^ x + 1) (a > 1), prove f (- x) = - f (x)
(3) Given that x satisfies the inequality 2 (log2) ^ 2-7log2 x + 3 ≤ 0, find the maximum and minimum of the function f (x) = log2 X / 2 · log2 X / 4

(1) Let g (x) = KX + B, then f (x) = f [g (x)] = 2 ^ (KX + b), and f ^ - 1 (x) = 1 / K (log2x-b); and point (2,1 / 4) is on the image of function f (x) and f ^ - 1 (x), so 1 / 4 = 2 ^ (2k + b) （1）,1/4=1/k(log22-b)…… (2) It can be concluded that 2K + B = - 2 (3),k+4b=4… ...

Why is the center of gravity of a triangle the center of gravity of a central triangle

Let AD and EF intersect at point P, because F and E are the midpoint of AB and AC, that is, EF is the median line of ABC, so EF is parallel to BC, so ape is similar to ABC, so PE / DC = AP / AD, so PE / DC = FP / BD, and DC = BD, so PE = FP, that is, DP is the midline of Fe, so point G is on the midline of Fe; similarly, point G is on the midline of De and DF, So G is the center of gravity of def

A math problem about function in senior one
Given that the function y = - 2x 2-12x-20 is translated according to the vector a = (h, K), so that the vertex is on the straight line x = 2, and the chord length obtained on the X axis is 6, find the analytic expression of the function after h, K and translation

Y = - 2x ^ 2-12x-20 = - 2 (x + 3) ^ 2-2, vertex (- 3, - 2)
The axis of symmetry of this function is x = - 3,
When the vertex is on the line x = 2, it shifts 2 + 3 = 5 units to the right, so h = 5
After translation, the vertex coordinates of the function are (2, m)
The obtained function is y = - 2 (X-2) ^ 2 + m.2
If the chord length obtained on the x-axis is 6, then it is 2. When y = 0, 0 = - 2 (X-2) ^ 2 + M,
That is, 2x ^ 2-8x + 8-m = 0, X1 + x2 = 4, x1x2 = (8-m) / 2
|X1-x2 | ^ 2 = (x1 + x2) ^ 2-4x1x2 = 16-4 * (8-m) / 2 = 36, M = 18
So the function after translation is y = - 2 (X-2) ^ 2 + 18,
So it's 18 + 2 = 20 units up, that's k = 20
So h = 5, k = 20, the analytic formula of the function is y = - 2 (X-2) ^ 2 + 18

The relationship between the monthly income of a car rental and loan company Y yuan and the monthly rent of each car X Yuan is y = - 50% x square + 162x-21000, so when the monthly income of each car and the rent of each car is how much yuan, the monthly income of the rental and loan company is the largest? What is the largest

∵ y = - X & sup2; / 50 + 162x-21000 = - 1 / 50 [x & sup2; - 50 * 162x] - 21000 = - 1 / 50 [x & sup2; - 8100 x + 4050 & sup2;] + 4050 & sup2; / 50-21000 = - 1 / 50 (x-4050) & sup2; + 328050-21000 = - 1 / 50 (x-4050) & sup2; + 307050 function opening downward, X ∈ (04050), monotone recurrence

More. Ha ha
1. An agricultural machinery company decided to support a village of a, B, C three different power diesel generators, a total of 10 (each at least one) and supporting the same model of 4 sets of pumps, 3 sets of 2 sets, each pump can pump water to irrigate farmland 1 mu per hour, now requires all diesel generators and matching pumps to work at the same time for 1 hour, irrigate farmland 32 mu. (1) set X type a diesel generators, The number of type B generators is y. The number of type C diesel generators is expressed by the formula containing X and Y. the functional relationship between Y and X is obtained

(1) (1) the number of C-type diesel generator is 10-x-y; (2) the number of C-type diesel generator is 10-x-y; (2) the number of C-type diesel generator is 10-x-y = (X-2), w = 130x + 120 (12-2x) + 100 (X-2) = - 10x + 1240. According to the inequality solution, we get: 3 ≤ x ≤ 5.5, ∵ x is a positive integer, ∵ x = 3,4

Given that the image of quadratic function y = x2 + kx-12 is shifted four unit lengths to the right, and the new image passes through the origin, then the value of K is ()
A. 4B. 3C. 2D. 1

∵ y = x2 + kx-12 = (x + 12K) 2-12-k24, ∵ parabola y = (x + 12K) 2-12-k24 is shifted to the right by 4 unit lengths. The analytic formula of the new parabola is y = (x + 12k-4) 2-12-k24. Substituting (0, 0) into (0 + 12k-4) 2-12-k24 = 0, the solution is k = 1

The cost price of a product per kilogram is 20 yuan, and its selling price is not lower than the cost price. When the selling price per kilogram is 50 yuan, its daily sales quantity is 100 kg. If the selling price per kilogram decreases (or increases) one yuan, the daily sales quantity increases (or decreases) 10 kg. Suppose the selling price per kilogram is x (yuan), the daily sales volume is y (kg), and the daily sales profit is w (yuan) (1) find the function analytic expression of Y with respect to X and write down the definition field of the function; (2) write down the function analytic expression of W with respect to X and the definition field of the function; (3) if the daily sales volume is 300 kg, please write down the daily sales profit directly

(1) Y = 100 + 10 (50-x), y = 600-10x, the domain is 20 ≤ x ≤ 50; (2) w = (600-10x) (x-20), w = - 10x2 + 800x-12000, the domain is 20 ≤ x ≤ 50; (7 points) (3) when the daily sales volume is 300kg, y = 600-10x = 300, the solution is: x = 30, substitute x = 30 into w = (600-10