Let $A = (4,-1)$, $B = (6,2)$, and $C = (-1,2)$. There exists a point $X$ and a constant $k$ such that for any point $P$,

\[PA^2 + PB^2 + PC^2 = 3PX^2 + k.\]Find the constant $k$.

I've tried, but got that k = 4x + 18 2/3, which is definitely wrong. I thought to take out the "4x", but its still wrong.

Why is no one helping :(

k=32

Step-by-step explanation:

Given the points:

The first step is to find the Centroid of the triangle.

Centroid, X

Next, let P be a point (x,y)

Using the distance formula,

On Substitution into:

Let us simplify the LHS first

Also, the Right Hand Side

Therefore:

the terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. a recursive formula for a geometric sequence with common ratio r is given by an=ran−1 a n = r a n − 1 for n≥2 n ≥ 2 .

step-by-step explanation:

the height of the surface increases from the center out to the sides of the road

it is c!