Expert Help with High School Geometry

Inner angles in polygons
A point of a polygon is called convex if it's inner angle is less than 180 degrees. Prove that in every simple polygon there is at least one convex point.

Distance between (2,5) and (9,1)?

Geometric probability - parallelograms
Inside a rhombus E with sides 10 unit and one interior angle less than 90 degree, there are 2 parallel (with E) parallelograms A and B, both can move freely and uniformly inside E but must keep parallel with E in moving. A is with base 8 unit and adjacent side 6 unit; while B is with base 5 unit and adjacent side 9 unit. If a point is chosen randomly in E, find the probability that the point lies inside A and B at the same time.

Jesse and three of his friends are playing Poker with 32 cards, The 32 cards are of every combination of the for patterns with the numbers 1, 7-13. In this game, each player takes five cards randomly.
1) Let X be the number of cards between the five Jesse got, with the number "1" on them. What is the probability function of X?
2) What is the probability that Jesse will have at least three cards with the same number?
3) During the evening, the four players played a number of rounds. In every round new cards were given out between the 32. Jesse decided he will stop playing after he will receive at lease three cards with the same number, in two games. What is the probability Jesse will play six games?
WIll someone please help me understand how to solve this question? In part c, is it a geometric probability based on part b ?

Ellipses Conics Inquiry
A carpenter wishes to construct an elliptical table top from a sheet 4ft by 8ft plywood to make a poker table for him and his budies. He will trace out the ellipse using the "thumbtack and string" method. What length of string should be used and how far apart should the tacks be located, if the ellipse is to be the largest possible that can be cut out of the plywood sheet?
I encountered this problem at my school and I'm having some trouble with it. Alright, so what I know already. The largest possible ellipse will have a minor axis of 4 feet and a major axis of 8 feet, so the a and b values of the ellipse equation will be 2 and 4, respectively.
The equation then will be: $x^2/16+y^2/4=1$
The vertices will be: (4,0), (-4,0), (0,2), (0,-2)
The combined distance from the foci to any point on the ellipse is the same, so the foci's coordinates can be (-f,0) and (f,0).

Decide whether each of these integers is congruent to 3 modulo 7.
66

Suppose that a and b are integers, ${\displaystyle a\equiv 11\left(b\text{mod}19\right),\text{and}b\equiv 3\left(b\text{mod}19\right)}$. Find the integer c with ${\displaystyle 0\le c\le 18}$ such that ${\displaystyle c\equiv 7a+3b\left(b\text{mod}19\right)}$

Suppose that a and b are integers, ${\displaystyle a\equiv 11\left(b\text{mod}19\right),\text{and}b\equiv 3\left(b\text{mod}19\right)}$. Find the integer c with ${\displaystyle 0\le c\le 18}$ such that ${\displaystyle c\equiv 8b\left(b\text{mod}19\right)}$

You're driving down the highway late one night at 20 m/s when a deer steps onto the road 35 m in front of you. Your reaction time before stepping on the brakes is 0.50 s, and the maximum deceleration of your car is ${\displaystyle 10\frac{m}{s^2}}$.
a. How much distance is between you and the deer when you come to a stop?
b. What is the maximum speed you could have and still not hit the deer?

Finding The Volume Of a Shape That Is Given By the Formula $3x^2+2y^2+z^2\le <!--\; \le \; -->6$
How can I find the volume of the shape that is given by the formula
$3x^2+2y^2+z^2\le <!--\; \le \; -->6$

Zero-modified distribution of a Geometric variable
Let X follow a Geometric distribution with probability of success 0.1, or $X\sim <!--\; \sim \; -->G(0.1)$. How do I calculate the probability that the zero-modified distribution of X with $f_X^M(0)=0.3$ is less than 4?

Finding volume triple integral
Find the volume of the solid enclosed on the outside of the sphere $r=2$ and on the inside by the surface $r=1+\cos <!--\; \; -->\mathrm{\varphi <!--\; \varphi \; -->}$.

Is there an intuitionist (i.e., constructive) proof of the infinitude of primes?

Let $\mathrm{△<!--\; △\; -->}ABC$ have $AB=15$, $BC=14$, and $CA=13.$. Let ${\mathrm{\ell <!--\; \ell \; -->}}_1$ be the angle bisector of $\mathrm{\angle <!--\; \angle \; -->}ABC$ and let ${\mathrm{\ell <!--\; \ell \; -->}}_2$ be the angle bisector of $\mathrm{\angle <!--\; \angle \; -->}BAC.$. If the projection of B onto ${\mathrm{\ell <!--\; \ell \; -->}}_2$ is Q and the projection of A onto ${\mathrm{\ell <!--\; \ell \; -->}}_1$ is P find PQ.

Perhaps a rather elementary question, but I simply couldn't figure out the calculations on this one. Say one takes a circle centeblack at the origin with radius $R$. He or she then proceeds to place $N$ circles with radius $r$ ($R>r$) on the larger circles circumference equidistantly, so every $2\mathrm{\pi <!--\; \pi \; -->}/N$ in the angular sense. What is then the relationship between $R$ and $r$ such that all neighboring circles exactly touch?
I've been trying to write down some equations with arc lengths and such for $N=4$, but I can't seem to get anything sensible out of it.

Distance between (-2,11) and (-11,15)?

Distance between (8,3,-5) and (6,1,2)?

Proving the Secant Angles in the Circle
How to prove the theorem of angles in the circle. Like this image here:

How prove that the angle X is the half of the sum of both angles' measurement of the Arc AC and Arc BD? This image here:

How prove that the angles A is the half of the difference of both angle measurements of Arc BC and Arc DE?

Distance between (-4,11) and (5,6)?

Use vectors to show that A(3,0,2), B(4,3,0), and C(8,1,-1) are vertices of a right triangle. At which vertex is the right angle?