Expert Help with High School Geometry

Find the radius of a circle with a circumference of $45\mathrm{\pi <!--\; \pi \; -->}$ centimeters

In triangle ABC, $AB=84,BC=112$, and $AC=98$. Angle B is bisected by line segment BE, with point E on AC. Angles ABE and CBE are similarly bisected by line segments BD and BF, respectively. What is the length of FC?

Suppose U and V are independent and follow the geometric distribution.
$p(k)=p(1-<!--\; -\; -->p{)}^k$ for $k=0,1,...$
Define the random variable $Z=U+V$
(a) Determine the joint probability mass function $P_{U,Z}(u,z)=Pr(U=u,Z=z)$
(b) Determine the conditional probability mass function for U given that $Z=n$

What is the expected distance between endpoints of n line segments of length 1 connected at random angles?
Start at the origin and take n line segments and connect them end to end each at random angles. What is the expected distance of the endpoint from the origin of the resulting path?
Clearly when $n=1$ the expected distance is 1. When $n=2$ we can find the expected distance by integrating
$\frac{1}{2\mathrm{\pi <!--\; \pi \; -->}}{\int <!--\; \int \; -->}_0^{2\mathrm{\pi <!--\; \pi \; -->}}2\sin <!--\; \; -->\left(\frac{x}{2}\right)dx=\frac{4}{\mathrm{\pi <!--\; \pi \; -->}}$
For $n=3$, it is easy to simulate and find the distance is approximately 1.58. For $n=4$ the simulated distance is 1.82 and for $n=5$ we get approximately 2.02.
Can one find a general formula for any n?

A geometric probability question
Find the probability of distance of two points ,which are selected in [0,a] closed interval, is less than ka $k<<!--\; <\; -->1$.

Conditional Expectation Property Geometric
I need to calculate $\mathbb{E}(X|Z)$ where X, Y are Geometric r.v. and $Z=X+Y$. The conditional distribution of $\mathbb{E}(X|Z)$ is $\frac{1}{n-<!--\; -\; -->1}$
So now, $\mathbb{E}(X|Z)=\underset{k=1}{\overset{n-<!--\; -\; -->1}{\sum <!--\; \sum \; -->}}\frac{k}{n-<!--\; -\; -->1}=\frac{n}{2}$. However, $\mathbb{E}(X)=\frac{1}{p}$. And by property of conditional Expectation $\mathbb{E}(\mathbb{E}(X|Z))=\mathbb{E}(X)$. So I get $\frac{n}{2}=\frac{1}{p}$. What am I doing wrong? How do I get $\mathbb{E}(\mathbb{E}(X|Z))=\frac{1}{p}$

An Olympiad Geometry problem with incenter configurations.
Let the inscribed circle of triangle ABC touches side BC at D ,side CA at E and side AB at F. Let G be the foot of the perpendicular from D to EF. Show that $\frac{FG}{EG}=\frac{BF}{CE}$.
So this problem is equivalent to proving similarity between $\mathrm{\Delta <!--\; \Delta \; -->}BFG$ and $\mathrm{\Delta <!--\; \Delta \; -->}CEG$.
I was able to prove that $\mathrm{\angle <!--\; \angle \; -->}GFB$= $\mathrm{\angle <!--\; \angle \; -->}GEC$ but after that, I hit a dead end. I found the point G pretty annoying as I couldn't apply any circle theorems to it.

What is the distance between (3,-5) and (2,-2)?

Conditional Geometric Probability
A certified public accountant (CPA) has found that nine of ten company audits contain substantial errors. If the CPA audits a series of company accounts, what is the probability that the first account containing substantial errors will occur on or after the third audited account?
The answer key tells me that it should be 0.01. Shouldn't it be 0.991 since $P(Y=3)=0.009$ and $1-<!--\; -\; -->0.009=0.991$?

Polygon of maximum area contained in compact, convex subset of ${\mathbb{R}}^2$?
Let H be a compact, convex subset of ${\mathbb{R}}^2.$. For a given $m\ge <!--\; \ge \; -->3,$, let $P_m$ be a polygon of maximum area which contained in H and has atmost m sides. Then ${\displaystyle \frac{Area(P_m)}{Area(H)}}\ge <!--\; \ge \; -->{\displaystyle \frac{m}{2\mathrm{\pi <!--\; \pi \; -->}}}\sin <!--\; \; -->\left({\displaystyle \frac{2\mathrm{\pi <!--\; \pi \; -->}}{m}}\right)$ and equality holds if and only if H is an ellipse.

What is the distance between (-9,3,1) and (1,5,2)?

Distance between (6,8,2) and (0,6,0)?

Why is the sum of probability is not equal to 1 in a geometric distribution?
A company has five employees on its health insurance plan. Each year, each employee independently has an 80% probability of no hospital admissions. If an employee requires one or more hospital admissions, the number of admissions is modeled by a geometric distribution with a mean of 1.50. The numbers of hospital admissions of different employees are mutually independent.
Each hospital admission costs 20,000.
Calculate the probability that the company’s total hospital costs in a year are less than 50,000.
The probability of each employee has at least one hospital admissions is $(0.2\cdot <!--\; \cdot \; -->1)/(1-<!--\; -\; -->2/3)=0.6$ instead of 0.2. Why is that?

Let $G$be an acyclic graph with $c$ components. Show that the number of edges of $G$ is $n-<!--\; -\; -->c$.

Formal definition of trigonometric functions
I want to define trigonometric function (say sine) formally with the definition that the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. But there is a problem with defining an angle (and the measure of an angle) without knowing trigonometric functions. Some people say that the measure of an angle is the ratio of the lenth of the arc to the lenght of the radius. But I don't know how to define an arc without trigonometric functions.
The solution may be to define sine and cosine with power series. I know this approach and it's fine, but I'm interested in classical definition.
So I came up with my own idea. Let $x\in <!--\; \in \; -->(0,90)$ and let P be a model of Hilbert's plane euclidian geometry, $\mathrm{\mu <!--\; \mu \; -->}$ is segments measure, $\mathrm{\nu <!--\; \nu \; -->}$ is angles measure such that the emasure of the right angle is 90, and $\mathrm{△<!--\; △\; -->}abc$ is a right triangle in which $\mathrm{\nu <!--\; \nu \; -->}(\mathrm{\angle <!--\; \angle \; -->}abc)=x$. Then $\sin <!--\; \; -->x=\frac{\mathrm{\mu <!--\; \mu \; -->}(ac)}{\mathrm{\mu <!--\; \mu \; -->}(ab)}$.
This definition requires proving many theorems (for instance existance of measures and triangle with given angles) and you have to prove that the definition doesn't depnd on the choice of the model, choice of the segments measure and choice of the right triangle.
The question is: Can we define angles and sine without referring to Hilbert' theory? Maybe it's possible to define measure of angles in euclidian model ${\mathbb{R}}^2$. I think the key part of the definition must be additivity of the measure, as it is in Hilbert's theory.

Let the vertices of a regular $72$-gon be colored red, blue, and green in equal parts. Show that there are $4$ vertices of each color such that the resulting monochromatic quadrilaterals formed are congruent to each other.

Distance between (5,6) and (8,8)?

Suppose $x_1,x_2,x_3\in <!--\; \in \; -->\mathbb{R}$. Prove that one of the $x_i$ must be greater than or equal to the average $\frac{1}{3}(x_1+x_2+x_3)$.