Expert Help with High School Geometry

I am trying to solve functional maximization problems. They are typically of the following form (where support of $\mathrm{\theta <!--\; \theta \; -->}$ is [0,1]):
${\displaystyle \int <!--\; \int \; -->[v(\mathrm{\theta <!--\; \theta \; -->},x(\mathrm{\theta <!--\; \theta \; -->}))+u(\mathrm{\theta <!--\; \theta \; -->},x(\mathrm{\theta <!--\; \theta \; -->}))-<!--\; -\; -->u_1(\mathrm{\theta <!--\; \theta \; -->},x(\mathrm{\theta <!--\; \theta \; -->}))(\frac{1-<!--\; -\; -->F(\mathrm{\theta <!--\; \theta \; -->})}{f(\mathrm{\theta <!--\; \theta \; -->})})]f(\mathrm{\theta <!--\; \theta \; -->})d\mathrm{\theta <!--\; \theta \; -->}}$
Now one way that was proposed to me was of point-wise maximization. That is you fix a $\mathrm{\theta <!--\; \theta \; -->}$ and then solve:
${\displaystyle argma{x}_{x(\mathrm{\theta <!--\; \theta \; -->})}v(\mathrm{\theta <!--\; \theta \; -->},x(\mathrm{\theta <!--\; \theta \; -->}))+u(\mathrm{\theta <!--\; \theta \; -->},x(\mathrm{\theta <!--\; \theta \; -->}))-<!--\; -\; -->u_1(\mathrm{\theta <!--\; \theta \; -->},x(\mathrm{\theta <!--\; \theta \; -->}))(\frac{1-<!--\; -\; -->F(\mathrm{\theta <!--\; \theta \; -->})}{f(\mathrm{\theta <!--\; \theta \; -->})})}$
Solving this problem would give me a number $x$ for each $\mathrm{\theta <!--\; \theta \; -->}$ and I will recover a function $x(\mathrm{\theta <!--\; \theta \; -->})$ that will maximize the original objective function.

I have two questions related to this:
1) Does such point-wise maximization always work?
2) What happens if rather than doing point-wise maximization I try and take the derivative of the objective function with respect to x(θ) and equating the first order condition to 0? Is this a legitimate way of solving the problem? Can someone show exactly what such a derivative would look like and how to compute it?

A circle has diameter AD of length 400.
B and C are points on the same arc of AD such that |AB|=|BC|=60.
What is the length |CD|?

How is the Pythagorean Theorem related to the Equation of a Circle
My teacher wants me to figure out how the Pythagorean Theorem and the Equation of a Circle are related. I can't figure this out because I view them as being two completely different things. I understand what the equation for a circle is, but how does that relate to the Pythagorean Theorem?

I am making a simple 3D printed mechanism that consists of a tab and housing. Both the tab and housing have nubs that interface with each other, preventing the tab from exiting the housing. The nubs are circular arcs with height 0.5mm and length 3.00mm. Fusion 360 calculated this for me to 4 decimal places, but I want to know the formula I can use, given the height and nub length, in order to find the exact value for the distance. This is necessary because I need the formula in order to use the output value for other aspects of the design.

Define the function
$f(a,b,c,\mathrm{\alpha <!--\; \alpha \; -->},\mathrm{\beta <!--\; \beta \; -->},\mathrm{\gamma <!--\; \gamma \; -->},x)=max(0,max((a+x)\mathrm{\alpha <!--\; \alpha \; -->},(b+x)\mathrm{\beta <!--\; \beta \; -->})-<!--\; -\; -->(c+x)\mathrm{\gamma <!--\; \gamma \; -->}),$
where
$a,b,c,\mathrm{\alpha <!--\; \alpha \; -->},\mathrm{\beta <!--\; \beta \; -->},\mathrm{\gamma <!--\; \gamma \; -->},x\in <!--\; \in \; -->[0,M].$
Is it true that for any
$\mathrm{\xi <!--\; \xi \; -->}=(a,b,c,\mathrm{\alpha <!--\; \alpha \; -->},\mathrm{\beta <!--\; \beta \; -->},\mathrm{\gamma <!--\; \gamma \; -->}),$
the maximum of $f(\mathrm{\xi <!--\; \xi \; -->},\cdot <!--\; \cdot \; -->)$ occurs either when $x=0$ or $x=M$?

I think the answer is yes, but I have trouble prooving it.
My argument is as follows:
Given any $\mathrm{\xi <!--\; \xi \; -->}$, $f(\mathrm{\xi <!--\; \xi \; -->},c)$ will be
$f(\mathrm{\xi <!--\; \xi \; -->},x)=\{\begin{array}{ll}0& \text{case A}\\ (a+x)\mathrm{\alpha <!--\; \alpha \; -->}-<!--\; -\; -->(c+x)\mathrm{\gamma <!--\; \gamma \; -->}& \text{case B}\\ (b+x)\mathrm{\beta <!--\; \beta \; -->}-<!--\; -\; -->(c+x)\mathrm{\gamma <!--\; \gamma \; -->}& \text{case C}\end{array}$
Hence, $f(\mathrm{\xi <!--\; \xi \; -->},x)$ is a linear function of $x$ in all three cases, and the result follows.

I think this argument works only if each case is independent of $x$, but this is not the case.

As, when $\mathrm{\xi <!--\; \xi \; -->}$ is given, judiciously choosing $x$ may put $f$ in another case.

What would be a right way to prove this result?

The following problem
$\underset{x,y}{max}f(x,y)=\log <!--\; \; -->x+\log <!--\; \; -->(x+y)-<!--\; -\; -->2x-<!--\; -\; -->3y$
subject to$x\ge <!--\; \ge \; -->0,y\ge <!--\; \ge \; -->0$

is a concave maximization problem, and thus can be numerically solved by convex optimization methods. The optimal solution may be $(x^{\ast <!--\; \ast \; -->},y^{\ast <!--\; \ast \; -->})=(1,0)$.

However, I want to derive the solution analytically. I tried the following approach, for example, by taking partial derivative,
$\frac{\mathrm{\partial <!--\; \partial \; -->}f}{\mathrm{\partial <!--\; \partial \; -->}x}=\frac{1}{x}+\frac{1}{x+y}-<!--\; -\; -->2$
$\frac{\mathrm{\partial <!--\; \partial \; -->}f}{\mathrm{\partial <!--\; \partial \; -->}x}=\frac{1}{x+y}-<!--\; -\; -->3$
And there exists no solution that makes both partial derivative to be 0.

Why is the Pythagorean Theorem not the Pythagorean Law?
Why do we call the Pythagorean Theorem a theorem, not a law?
As far as I know, we call a theorem a theorem because though it's reliable in every observable case, its truthfulness cannot be proven for every case. However I've looked and it seems as though we (as the human race) have very extensive proofs of the Pythagorean Theorem, considering every case. Why is it that it hasn't made the transition from theorem to law? Is my terminology confused? Is it not as proven as I think it is? Is there some other condition for being a law that I don't know about?

Earlier last week I realized I needed to ship a large volume of things domestically. Of course, I decided that I wanted to do so as cheaply as possible.

I first looked at USPS standard post rates. I noted that if the "combined length and girth" of a box exceeds 108", then the package becomes significantly more expensive, as it uses the "oversized" price bracket, so I resolved not to exceed this size limit. The length of a box is defined as its longest dimension, and its girth is defined as the perimeter around the rectangle to which the length is orthogonal – in other words, the girth is $2w+2h$.

However, because I have much to ship, I also would like to maximize the volume of such a box while not exceeding this limit. This turns into a maximization problem with a constraint:

Maximize $V(x)=lwh$, while satisfying $108>=l+2w+2h$.

This seems pretty simple, but I'm not sure how to handle the three nominally independent variables $l$, $w$, and $h$. Because I did not want to wait or rely on my faulty math skills, I ultimately asked Wolfram|Alpha to solve this for me, yielding a box of size 36"x18"x18". This makes intuitive sense, to a degree, but I would like a hint as to how I could have proceeded to solve this on my own.

Let $AB$ be an arc of a circle. Tangents are drawn at $A$ and $B$ to meet at $C$. Let $M$ be the midpoint of arc $AB$. Tangent drawn at $M$ meet $AC$ and $BC$ at $D$, $E$ respectively. Evaluate
$\underset{AB\to <!--\; \to \; -->0}{lim}\frac{\mathrm{\Delta <!--\; \Delta \; -->}ABC}{\mathrm{\Delta <!--\; \Delta \; -->}DEC}$
I don't think evaluating the limit will be a problem. But how do I find the areas of the triangles and in what parameters?

I Have read that for a matrix of reals $Y$ and a p.s.d matrix $B$ that the

Maximum of $f(Y)=Tr(Y^{T}BY)$ subject to $Y^{T}Y=I$ is achieved when $span(Y)$ equals the span of the first $d$ eigen-vectors of $B$.

What is the reasoning behind this eigen-solution leading to the maximum?

Euclidean norm vs pythagorean theorem?
I recently stumbled upon an Euclidian norm. First I thought there are the powers and square root to deal with possible negative values (like in Standard deviation formula) but then I realized, the final number (sum of squares and square root of it) is not same as sum of absolute numbers.
So then I noticed it is more related to Pythagorean theorem (Euclidian distance).
But if looked in google for PT in 3d space i found the formula like
$(a^2+b^2+c^2{)}^{\frac{1}{2}}$
But in the Euclidian norm it is
$SQRT(a^2+b^2+c^2)$
How comes, it is different?
Second question, how comes that the theorem works for any count of dimensions?

I'd like to compute
$A^{\ast <!--\; \ast \; -->}=\underset{\begin{array}{c}A\in <!--\; \in \; -->{\mathbb{R}}^{d\times <!--\; \times \; -->k}\\ A^{T}A=I\end{array}}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}}<!--\; \; -->det(A^T\mathrm{\Lambda <!--\; \Lambda \; -->}A)$
where $k\le <!--\; \le \; -->d$, $\mathrm{\Lambda <!--\; \Lambda \; -->}=\mathrm{diag}<!--\; \; -->({\mathrm{\lambda <!--\; \lambda \; -->}}_1,\dots <!--\; \dots \; -->,{\mathrm{\lambda <!--\; \lambda \; -->}}_d)$ and ${\mathrm{\lambda <!--\; \lambda \; -->}}_1\ge <!--\; \ge \; -->\cdots <!--\; \cdots \; -->\ge <!--\; \ge \; -->{\mathrm{\lambda <!--\; \lambda \; -->}}_d\ge <!--\; \ge \; -->0$.

I strongly suspect that $A^{\ast <!--\; \ast \; -->}=[{\mathrm{ϵ<!--\; ϵ\; -->}}_1\cdots <!--\; \cdots \; -->{\mathrm{ϵ<!--\; ϵ\; -->}}_k]$, but after various attempts to prove it I gave up.

Equation of the ellipse between 2 points (its vertices)
Start point and end point:
$A(x,y)B(x,y)$
Ellipse width:
$w=2.5$
Midpoint or center of the ellipse:
$M=(\frac{A_x+B_x}{2},\frac{A_y+B_y}{2})$
The angle (that would be the angle between points A and B):
$\mathrm{\theta <!--\; \theta \; -->}=?$
Parametric equation of the ellipse with rotation:
$x=a\ast <!--\; \ast \; -->cos(t)\ast <!--\; \ast \; -->cos(\mathrm{\theta <!--\; \theta \; -->})+b\ast <!--\; \ast \; -->sin(t)\ast <!--\; \ast \; -->sin(\mathrm{\theta <!--\; \theta \; -->})$
$y=b\ast <!--\; \ast \; -->sin(t)\ast <!--\; \ast \; -->cos(\mathrm{\theta <!--\; \theta \; -->})-<!--\; -\; -->a\ast <!--\; \ast \; -->cos(t)\ast <!--\; \ast \; -->sin(\mathrm{\theta <!--\; \theta \; -->})$

Pythagorean theorem
We can make a square into four equal squares. Fine, if we want to make into five.. Then there is a problem. Please discuss, How to make five squares from a single square by using a Pythagorean theorem. Is there any other way to make five squares from one square without using Pythagorean theorem? Please discuss. Thanking you, KKRG

Suppose, $2x\cos <!--\; \; -->\mathrm{\alpha <!--\; \alpha \; -->}-<!--\; -\; -->3y\sin <!--\; \; -->\mathrm{\alpha <!--\; \alpha \; -->}=6$ is equation of a variable straight line. From two point $A(\sqrt{5},0$ and $B(-<!--\; -\; -->\sqrt{5},0)$ foot of the altitude on that straight line is P and Q respectively. Show that the product of the length of two line segment AP and BQ is free of $\mathrm{\alpha <!--\; \alpha \; -->}$ .

Consider a circle with a circumference of n. On this circle, I define two arcs of length $k<n$, $A_1$ and $A_2$. The centres of the two arcs are uniformly distributed on the circle.
Let ${\mathrm{\Omega <!--\; \Omega \; -->}}_1=A_1\setminus <!--\; \setminus \; -->A_2$ and ${\mathrm{\Omega <!--\; \Omega \; -->}}_2=A_2\setminus <!--\; \setminus \; -->A_1$ such that the length of ${\mathrm{\Omega <!--\; \Omega \; -->}}_1\cup <!--\; \cup \; -->{\mathrm{\Omega <!--\; \Omega \; -->}}_2$ is $2k$ minus the overlapping part of the two arcs.
What is the expectation of the length of ${\mathrm{\Omega <!--\; \Omega \; -->}}_1\cup <!--\; \cup \; -->{\mathrm{\Omega <!--\; \Omega \; -->}}_2$?

Show that the points $(9,1),(7,9),(-<!--\; -\; -->2,12),(6,10)$ are concyclic.

$arg(\frac{z-<!--\; -\; -->a}{z-<!--\; -\; -->b})=c$
My understanding is as follows. The angle c between the lines za and zb is constant. za and zb meet at z and the angle between the lines perpendicular to za and zb is 2c, which is only the case if z lies on a circle passing through z, a and b with centre where the lines perpendicular to za and zb intersect. So the locus is an arc of a circle through a and b except a and b (because $arg(\frac{z-<!--\; -\; -->a}{z-<!--\; -\; -->b})$ is undefined there). Is this reasoning correct?

Is there another way the statement could this be expressed (e.g. in terms of moduli)? Also, what is an efficient way of finding the centre of the circle?

I have the following problem. Given this function
$E[\mathrm{\pi <!--\; \pi \; -->}]=(1-<!--\; -\; -->r)[\mathrm{\alpha <!--\; \alpha \; -->}b-<!--\; -\; -->(1-<!--\; -\; -->p)C-<!--\; -\; -->K]+T$
I would like to find the maximum w.r.t. $r$ given this constraint:
$U=(1-<!--\; -\; -->r)b-<!--\; -\; -->T\ge <!--\; \ge \; -->0$
It is an economic problem that I am formalizing, but this is not relevant to its solution, I am only interested in the mathematical resolution of the problem. I provide some background. Our variable $r$ is a number between 0 and 1, $p$ is some probability, $\mathrm{\alpha <!--\; \alpha \; -->},b,C,K$ and $T$ are all positive constants. If it is useful for the resolution of the problem, we can also assume that $\mathrm{\alpha <!--\; \alpha \; -->}$ is between 0 and 1. An important assumption (namely, assumption $\mathrm{\&<!--\; \&\; -->}$) is that $(1-<!--\; -\; -->p)C+K>\mathrm{\alpha <!--\; \alpha \; -->}b$. Readers used to economics may recognize that the objective function is a sort of expected profit and the constraint is a sort of expected utility. I tried with Kuhn-Tucker, but with miserable results since deriving w.r.t. $r$ does not yield any expression with $r$.

Now I'm following a more intuitive approach. I start by assuming that $U=0$ is the constraint, then I can find an expression for $r$ from the constraint and I substitute it in the target function. After easy steps, I get this
$T\frac{\mathrm{\alpha <!--\; \alpha \; -->}b-<!--\; -\; -->(1-<!--\; -\; -->p)C-<!--\; -\; -->K}{b}+T$
At this point, I can use assumption $\mathrm{\&<!--\; \&\; -->}$ to conclude that the first $T$ above will be negative, but I'm not able to conclude whether it will be lower, equal or higher to/than the second $T$ because it is multiplied by some constant.

At this point I'm stuck. I do not know if my approach can work. Could you please suggest a nicer way to proceed? I am on the right track or it's a dead end?