Expert Help with High School Geometry

In $\mathrm{\Delta <!--\; \Delta \; -->}ABC,AC>AB.$. The internal angle bisector of $\mathrm{\angle <!--\; \angle \; -->}A$ meets BC at D, and E is the foot of the perpendicular from B onto AD. Suppose $AB=5,BE=4$ and $AE=3$. Find $(\frac{AC+AB}{AC-<!--\; -\; -->AB})ED$.

Distance between (3,-2,-12) and (5,-8,-16)?

What is this geometric probability
In a circle of radius R two points are chosen at random(the points can be anywhere, either within the circle or on the boundary). For a fixed number c, lying between 0 and R, what is the probability that the distance between the two points will not exceed c?

Distance between (2,5) and (5,-3)?

Greatest distance one point can have from a vertice of a square given following conditions
A point P lies in the same plane as a given square of side 1.Let the vertices of the square,taken counterclockwise,be A,B,C, and D.Also,let the distances from P to A,B, and C, respectively, be u,v and w.
What is the greatest distance that P can be from D if $u^2+v^2=w^2$?
Some thoughts I had:
1) Given a pair of vertices I could construct an ellipse with P as a point on the ellipse.
2) From the equality $u^2+v^2=w^2$. I think that I have to consider the case where the angle between u and v is ${90}^{\circ <!--\; \circ \; -->}$. In this case I would have $w=1$ and $PD<<!--\; <\; -->2$.
That being said,I still fail to come at a concrete solution of the problem,it might be that none of my thoughts are right...

What is the distance between (8,5) and (1,2)?

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

The inscribed angle theorem states that $\mathrm{\angle <!--\; \angle \; -->}O=2\times <!--\; \times \; -->\mathrm{\angle <!--\; \angle \; -->}B$. The theorem is true for when point B is located between points A and C relative to the perimeter. But what would happen if B was located exactly at A or C? Angle B would obviously equal 0 and so would the length of one of it's legs, but my question is, is this transition discrete or continuous? In other words, as the length of either line BA or BC approaches 0, does angle B also approach 0? Or is it unaffected?

Distance between (3,-1,1) and (3,2,1)?

How to find sum of infinite geometric series with coefficient?
Given this series, $p+p(1-<!--\; -\; -->p{)}^3+p(1-<!--\; -\; -->p{)}^6+p(1-<!--\; -\; -->p{)}^9+...$
This is an infinite geometric series with ratio less than 1 since it's probability.
$\underset{n=0}{\overset{\mathrm{\infty <!--\; \infty \; -->}}{\sum <!--\; \sum \; -->}}p(1-<!--\; -\; -->p{)}^{3n}$
Can you use geometric series sum formula? Is it $p/(1-<!--\; -\; -->(1-<!--\; -\; -->p{)}^3)$?
How do you deal with 3 that's in front of n?

Pyramid SABC has right triangular base ABC, with $\mathrm{\angle <!--\; \angle \; -->}ABC={90}^{\circ <!--\; \circ \; -->}$. Sides $AB=\sqrt{3},BC=3$. Lateral lengths are equal and are equal to 2. Find the angle created by lateral length and the base.

Finding volume of a solid about a vertical line
How can I calculate the volume of the solid generated by rotating the region bounded by
$y=x^2-<!--\; -\; -->6x+10$
and $y=-<!--\; -\; -->x^2+4x+2$.
About the line $x=4$

Proof of angle Bisector Theorem by Vectors
For $\mathrm{\Delta <!--\; \Delta \; -->}ABC$, let AD is its internal angle bisector, then we know that $\frac{AB}{AC}=\frac{BD}{CD}$ and I know that it can be proved by geometry by drawing a line through C parallel to AD and extend AB so that it cuts new line and then use various angles and concept of similar triangle to prove our result. But I was wondering if we can prove it with the help of vectors as well?
If we take position vectors of A,B and C as $\stackrel{\to <!--\; \to \; -->}{0},\stackrel{\to <!--\; \to \; -->}{b}$ and $\stackrel{\to <!--\; \to \; -->}{c}$ respectively , then $\stackrel{\to <!--\; \to \; -->}{AD}$ can be taken as $\mathrm{\lambda <!--\; \lambda \; -->}(\stackrel{^<!--\; ^\; -->}{b}+\stackrel{^<!--\; ^\; -->}{c})$. Could someone help me to proceed after that or provide some alternate approach through vectors?

Distance between (3,5) and (-5,13)?

100 Gambles of Roulette
I am stuck on the following problem: You go to Las Vegas with $1000 and play roulette 100 times by betting $10 on red each time. Compute the probability of losing more than $100. Hint: Each bet you have chance 18/38 of winning $10 and chance 20/38 of losing $10.
So essentially this is recurring 100 times. Now each time you do it you have an 18/38 chance of winning and 20/38 chance of losing. You have a greater chance of losing than winning, so based on speculation you need to lose 56/100 in order to lose more than $100, to counterbalance your winnings . How would I set up an equation to solve for this?

Distance between (3,-2,-2) and (5,-4,-6)?

Conditional Probability in Geometric Distribution
Ms. A is taking driving test to get her driving license. The probability of passing a test is 0.3 which remains same no matter how many times she takes the test. Let X is a random variable which is the number of tests she takes to get her license.
1) Use conditional probability rule to determine her chance of getting the license on the 6th test starting from her first attempt considering it is known that she has already failed on the first 4 tests.
2) Though one can take the driving test 6 times within a six-month period, he/she can take tests until getting licensed. How many tests is Ms. A expected to take to get the license? What is the standard deviation of the number of tests she takes to get licensed?

Suppose 10 patients are to be tested for a blood disease and that the test in guaranteed to detect the disease. Furthermore, suppose that the probability that a patient has the disease is 0.02 and that presence of the disease in any patient is independent of all other patients. One option is to test all patients individually and that would require 10 tests. Another option is to combine blood samples from the 10 patients and test the combined sample. If the test is negative, then all patients are declared disease free and only 1 test is required. If, however, the combined sample tests positive, then all 10 patients are tested individually, requiring a total of 11 tests. In the latter option derive the probability distribution of , the number of tests, and determine Expected Value?
Among all the distribution models, it seems that geometric distribution seems best fit. But, when I use geometric distribution pmf and its' $EV=1/p$ , I get unreasonable answers.

Distance between (2,3) and (3,0)?

In $\mathrm{△<!--\; △\; -->}ABC$, let D be a point on BC such that AD bisects $\mathrm{\angle <!--\; \angle \; -->}A$. If $AD=6$, $BD=4$ and $DC=3$, then find AB.