Expert Help with High School Geometry

Pythagorean theorem and orthocenter
Point H is the orthocenter of $\mathrm{△<!--\; △\; -->}ABC$. Let AD and CM be the altitudes. Show that $B{M}^2-<!--\; -\; -->B{D}^2=H{D}^2-<!--\; -\; -->H{M}^2$.
By Pythagorean theorem: 1) △$\mathrm{△<!--\; △\; -->}BCM\to <!--\; \to \; -->B{M}^2=B{C}^2-<!--\; -\; -->C{M}^2$ and 2) $\mathrm{△<!--\; △\; -->}ABD\to <!--\; \to \; -->B{D}^2-<!--\; -\; -->A{B}^2-<!--\; -\; -->A{D}^2$ How to approach the problem further?

In $\mathrm{△<!--\; △\; -->}ABC,$ , $B=(-<!--\; -\; -->7,1),C=(-<!--\; -\; -->5,-<!--\; -\; -->5)$
Centroid $G=(h,k)$ and orthocentre $H=(3h,3k)$ . If $A=(1,Y_A)$ then find $Y_A$

I try to maximize a quadratic function on box constraints as shown below
$\begin{array}{l}max{x}^{T}Qx\\ s.t.a\le <!--\; \le \; -->x_i\le <!--\; \le \; -->b\end{array}$
The length of vector $x$ is approximately 2000 and $Q$ is symmetric. I am wondering if there exists a solver (like Cplex or Gurobi) which can solve this maximization problem efficiently.

Pythagorean Theorem - Incomplete proof
The image below is supposed to prove Pythagorean Theorem (without words). However, I see that it is assuming that the square of side c can be inscribed in the square of side a+b and would result in the 4 triangles of equal area. This fact is only assumed, so would the proof be considered incomplete?

"A 10-meter length of wire is available for making a circle and a square. How should the wire be distributed between the two shapes to maximize the sum of the enclosed areas?"

Here's what I have:
$Are{a}_c=\mathrm{\pi <!--\; \pi \; -->}{r}^2$
$Are{a}_s=4r^2$
So, I'm thinking that I need to find the maximized radius size to figure everything else out.
$Are{a}_c+Are{a}_s=10$
$(\mathrm{\pi <!--\; \pi \; -->}{r}^2)+(4r^2)=10$
$\frac{\mathrm{d}}{\mathrm{d}<!--\; \; -->r}[(\mathrm{\pi <!--\; \pi \; -->}{r}^2)+(4r^2)-<!--\; -\; -->10]=2\mathrm{\pi <!--\; \pi \; -->}r+8r$
But here's my dilemma; if I take the derivative of that and solve for r, it comes out 0. So I'm not sure where I'm going wrong. Any advice?

The things we know, usually minimization of a convex function, unique solution will exist.

My question is, maximization of a strictly convex function, will that give an unique maximum? If so how we can proof that. The objective function looks like below.
max $A{e}^{-<!--\; -\; -->(d_0+d_1)}+B{e}^{-<!--\; -\; -->(d_0+d_1+d_2)}+C$
A,B and C are constants and $d_0,d_1,d_2$ are euclidean distances.

Is it possible to solve the value of lambda that maximizes the following equation analytically?:
$\frac{1-<!--\; -\; -->e^{-<!--\; -\; -->30\mathrm{\lambda <!--\; \lambda \; -->}}}{30\mathrm{\lambda <!--\; \lambda \; -->}}-<!--\; -\; -->e^{-<!--\; -\; -->30\mathrm{\lambda <!--\; \lambda \; -->}}$
So then the derivative is equal to:
$\frac{e^{-<!--\; -\; -->30\mathrm{\lambda <!--\; \lambda \; -->}}}{\mathrm{\lambda <!--\; \lambda \; -->}}-<!--\; -\; -->\frac{30}{900{\mathrm{\lambda <!--\; \lambda \; -->}}^2}+\frac{30e^{-<!--\; -\; -->30\mathrm{\lambda <!--\; \lambda \; -->}}}{900{\mathrm{\lambda <!--\; \lambda \; -->}}^2}+30e^{-<!--\; -\; -->30\mathrm{\lambda <!--\; \lambda \; -->}}=0$
Is it possible to get an analytical solution for lambda or does it need to be solved numerically?

Pythagorean theorem
The Pythagorean theorem is typically written as
$a^2+b^2=c^2$
where a and b are the two shorter sides (legs) of a right triangle and c is its longest side (the hypotenuse).
Given the exponent powers $c^{2\times <!--\; \times \; -->1/2}=c$, why cant I express "$a=c-<!--\; -\; -->b$".
The correct answer given was $\sqrt{c^2-<!--\; -\; -->b^2}$

If ($x,y,z$) be the lengths of perpendiculars from any interior point P of a triangle $ABC$ on sides $BC$,$CA$ and $AB$ respectively then find the minimum value of :
$x^2+y^2+z^2$
The sides of triangle being $a,b,c$.

I thought of using Lagrange's method of multipliers but am not able to find another function in terms of $x,y,z$ and $a,b,c$Any help will be appreciated. Thanks.

Take the Viviani curve intersection of a sphere and a cylinder. Analytically explain what happens to the intersection curve if you keep the radius of the sphere constant, fix one side of the cylinder at the origin, and change the radius of the cylinder. Especially, discuss the cases when the radius of the cylinder goes to 0 and $\mathrm{\infty <!--\; \infty \; -->}$

Focuses of the hyperbola $F_1(4,2),F_2(-<!--\; -\; -->1,-<!--\; -\; -->10)$ and the tangent equation $3x+4y-<!--\; -\; -->5=0$ are given. Find the semiaxes of the hyperbola.
I’ve tried using the definition of hyperbola and the equation for the tangent line to the quadratic curve to form some sort of system of equations, but it didn’t work out.

Determine a so that the intersection line between the planes
$P_1:2x+ay-<!--\; -\; -->z=3$
$P_2:x-<!--\; -\; -->2y+az=5$
are parallel to the plane $P_3:2x+y+z=2$

If we have that the contours of a response surface are elliptical and the response is given by the following function:
$\exp <!--\; \; -->(-<!--\; -\; -->(w^2+\frac{1}{4}{l}^2-<!--\; -\; -->\frac{1}{4}\cdot <!--\; \cdot \; -->w\cdot <!--\; \cdot \; -->l))$
then if we maximize this function w.r.t l holding w fixed at 1/2.
And if we call the maximizer l-star, then holding l-star fixed, maximize over w. How to show that the overall max isn't achieved?
My approach: I got the partial of the above function w.r.t. l and then tried to evaluate it at 1/2, but got stuck. It most likely will involve some analysis of Hessians.

Can the Pythagorean Theorem be extended like this?
What is the significance, if any, of the fact that
$3^3$ + $4^3$ + $5^3$ = $6^3$
?
How curious this is. This would be like the Pythagorean Theorem exploding into a higher dimension, on steroids.

I need help solving this following equation, a Lagrangian problem that I encountered during my studies in principal component analysis (PCA).

One should maximize the variance with respect to the first principal component

Following is the case: We know that $W={\mathbf{v}}_1^T{\mathbf{S}\mathbf{v}}_1$ (eq 1) and that ${\mathbf{v}}_{\mathbf{1}}$ is an orthonormal basis.

Therefore $||{\mathbf{v}}_1|{|}^2={\mathbf{v}}_1^T{\mathbf{v}}_1=1$ (eq 2). The maximization of $W$ under this constraint can be done by introducing the Lagrangian multiplier $\mathrm{\lambda <!--\; \lambda \; -->}$ and maximizing the Lagrangian:
$\mathcal{L}=W+\mathrm{\lambda <!--\; \lambda \; -->}(1-<!--\; -\; -->||{\mathbf{v}}_1|{|}^2)={\mathbf{v}}_1^T(\mathbf{S}-<!--\; -\; -->\mathrm{\lambda <!--\; \lambda \; -->}\mathbf{I}){\mathbf{v}}_1+\mathrm{\lambda <!--\; \lambda \; -->}$ (eq 3)
So expanding $W$ using eq. 1, and the laws of distributivity we end up with eq 3.
Next, taking the derivative with respect to vector and $\mathrm{\lambda <!--\; \lambda \; -->}$ we obtain:
$\frac{\mathrm{\partial <!--\; \partial \; -->}}{\mathrm{\partial <!--\; \partial \; -->}\mathrm{\lambda <!--\; \lambda \; -->}}\mathcal{L}=1-<!--\; -\; -->{\mathbf{v}}_1^T{\mathbf{v}}_1=0$
${\mathrm{\nabla <!--\; \nabla \; -->}}_{\mathbf{V}}{}_1\mathcal{L}=(\mathbf{S}-<!--\; -\; -->\mathrm{\lambda <!--\; \lambda \; -->}\mathbf{I}){\mathbf{v}}_1=0$
This can later be shown to be an eigenvalue problem. Can anyone explain to me why the derivatives look like that? I can't seem to figure it out.