Elementary Geometry Word Problems Here!

Prove that angle bisectors of a triangle are concurrent using vectors. Also, find the position vector of the point of concurrency in terms of position vectors of the vertices.

I solved this without using vectors to get some idea. I am not sure how to prove it using vectors. I don't want to use vector equations for straight lines and then find the point of concurrency. That's like solving using coordinate geometry.

Let $\stackrel{\to <!--\; \to \; -->}{a},\stackrel{\to <!--\; \to \; -->}{b},\stackrel{\to <!--\; \to \; -->}{c}$ represent the sides $A,B,C$ respectively.

The angle bisectors are along ${\displaystyle \frac{\stackrel{\to <!--\; \to \; -->}{a}}{|\stackrel{\to <!--\; \to \; -->}{a}|}}+{\displaystyle \frac{\stackrel{\to <!--\; \to \; -->}{b}}{|\stackrel{\to <!--\; \to \; -->}{b}|}},{\displaystyle \frac{\stackrel{\to <!--\; \to \; -->}{b}}{|\stackrel{\to <!--\; \to \; -->}{b}|}}+{\displaystyle \frac{\stackrel{\to <!--\; \to \; -->}{c}}{|\stackrel{\to <!--\; \to \; -->}{c}|}},{\displaystyle \frac{\stackrel{\to <!--\; \to \; -->}{c}}{|\stackrel{\to <!--\; \to \; -->}{c}|}}+{\displaystyle \frac{\stackrel{\to <!--\; \to \; -->}{a}}{|\stackrel{\to <!--\; \to \; -->}{a}|}}$

Let the sides $AB,BC,CA$ be $x,y,z$. Let $AD$ be one of the angular bisector.
$\frac{BD}{CD}=\frac{x}{z}$
Hence
$D=\frac{x\stackrel{\to <!--\; \to \; -->}{c}+z\stackrel{\to <!--\; \to \; -->}{b}}{x+z}$
What should be the next step? Or is there a better method?

For reference: In the drawing, T is the point of tangency, $LN||AT$, $OH=4$ and $L{N}^2+A{M}^2=164$ . Calculate HN.

This might be an easy question, but since i'm new to solid shapes, i couldn't solve it. $A=(7,1,3)B=(5,1,2)C=(4,-<!--\; -\; -->2,3)D=(6,m,n)$ I need to find m and n so that segments BD and AC have the smae midpoint.

I'm trying to get an overarching understanding of the components of mathematical systems so that in my self study of each category of math I can break them down by their unique aspects, i.e. the operators they use, the major concepts they deal with (i.e. how calculus is about "change"), etc.

As far as my experience with formal math terminology goes, im rather weak, and i get utterly confused by the technicality required in formal definitions.

As a good starting point, I'd like to better understand what the difference is between an axiom, a theorem and a postulate. At my current level of knowledge i would use them interchangeably (lol), however I'm sure one is founded upon the others.

If someone could explain the logical hierarchy/relation between these three it would be greatly appreciated.

Is it true that:

-an inner product satisfies the properties of a norm if and only if the norm satisfies the parallelogram equality

-a norm can be induced by a metric if and only if the metric satisfies $d(x+a,y+a)=d(x,y)$ and $d(ax,ay)=ad(x,y)$

or are the implications one way?

I've looked all over and I can't find a good proof of why the diagonals of a rhombus should intersect at right angles. I can intuitively see its true, just by drawing rhombuses, but I'm trying to prove that the slopes of the diagonals are negative reciprocals and its not working out.

I'm defining my rhombus as follows: $[(0,0),(a,0),(b,c),(a+b,c)]$

I've managed to figure out that $c=\sqrt{a^2-<!--\; -\; -->b^2}$ and that the slopes of the diagonals are $\frac{\sqrt{a^2-<!--\; -\; -->b^2}}{a+b}$ and $\frac{-<!--\; -\; -->\sqrt{a^2-<!--\; -\; -->b^2}}{a-<!--\; -\; -->b}$

What I can't figure out is how they can be negative reciprocals of one another.

I mean to say that I could not find the algebraic proof. I've seen and understand the geometric proof, but I needed help translating it into coordinate form.

I'm given that the distance between two points in the Beltrami-Klein model is
$d(XY)=\frac{1}{2}ln(\frac{\stackrel{¯<!--\; ¯\; -->}{XQ}\cdot <!--\; \cdot \; -->\stackrel{¯<!--\; ¯\; -->}{YP}}{\stackrel{¯<!--\; ¯\; -->}{XP}\cdot <!--\; \cdot \; -->\stackrel{¯<!--\; ¯\; -->}{YQ}})$
where P and Q are ideal points lying on the boundary of the unit disc, and $\stackrel{¯<!--\; ¯\; -->}{XQ}$ denotes the standard Euclidean distance between a point X inside the unit disc and an ideal point Q.
Given the ideal points $P=(0,1),Q=(0,-<!--\; -\; -->1)$ , nad points $A=(0,0)$ and $B=(0,\frac{1}{2})$ . I am asked to find the midpoint $M=(0,m)$ between A and B.

I'm trying to prove some properties of sequence spaces. I already know that the space $l^{\mathrm{\infty <!--\; \infty \; -->}}$ of all bounded sequences isn't an inner product space, isn't separable but it is complete with sup norm.
The space c of all convergent sequences is also complete, as a closed subspace of $l^{\mathrm{\infty <!--\; \infty \; -->}}$, and so is $c_0$ - the space off all sequences with entries equal to zero from some point on.
I also know that both $c$ and $c_0$ are separable.
My question is - are $c$ and $c_0$ inner product spaces. Should I look for an example of sequences which don't satisfy the parallelogram law.

Please let me know the formula for the coordinate of the midpoint of 2 points in spherical coordinate system.

I have been recently studying a C.G. Hempel's article on mathematical truth and pointed out his following quotation: "Every concept of mathematics can be defined by means of Peano's three primitives,and every proposition of mathematics can be deduced from the five postulates enriched by the definitions of the non-primitive terms".

I was wondering if it is possible for someone to make a scheme illustrating the sequence of the derivation of the whole theory of mathematics being derived by these postulates.Possibly starting from natural numbers?(notice that Hempel excludes geometry)

Prove $|x+y{|}^2+|x-<!--\; -\; -->y{|}^2=2|x{|}^2+2|y{|}^2$ if $x,y\in <!--\; \in \; -->{\mathbb{R}}^k$.
Interpret this geometrically, as a statement about parallelograms.
I've shown that the expression given equates to $x\cdot <!--\; \cdot \; -->x+2x\cdot <!--\; \cdot \; -->y+y\cdot <!--\; \cdot \; -->y+x\cdot <!--\; \cdot \; -->x-<!--\; -\; -->2x\cdot <!--\; \cdot \; -->y+y\cdot <!--\; \cdot \; -->y$. But what does this have to do with parallelograms?

The internal bisectors of the angles of a triangle ABC meet the sides in D,E,and F.Show that area of the triangle DEF is equal to $\frac{2\mathrm{\Delta <!--\; \Delta \; -->}\times <!--\; \times \; -->abc}{(b+c)(c+a)(a+b)}$,here $\mathrm{\Delta <!--\; \Delta \; -->}$ is area of triangle ABC

If I choose B as origin,C as(a,0),a is side length BC.what should i take coordinates of A,can i get answer with this approach or any better approach?

Finding the slopes of internal or external angle bisectors, when given the slopes of the sides of a triangle, it's a simple question, the one you can easily find in any textbook of analytic geometry.

But the reverse problem : finding the slopes of the sides of a triangle, when only given the slopes of internal angle bisectors. Is it possible? Is there a straightforward formula for that?

The way I view Euclid's postulates are as follows:

A line segment can be made between any two points on surface A.

A line segment can be continued in its direction infinitely on surface A.

Any line segment can form the diameter of a circle on surface A.

The result of an isometry upon a figure containing a right angle preserves the right angle as a right angle on surface A.

If a straight line falling on two straight lines make the interior angles on the same side less than 180 degrees in total, the two straight lines will eventually intersect on the side where the sum of the angles is less than l80 degrees.

The way I view the five postulates is simple. Each postulate defines some quality a surface has.

For instance, 2 seems to define whether a surface is infinite/looped or finite/bounded, 3 seems to force a surface to be circular (or a union of circular subsets), and 5 I believe change the constant curvature of a surface (wether it is 0 or nonzero).

I want to determine what "quality" 1 and 4 define in the context of the surface itself. 1 seems to imply discontinuity vs continuity, and I think 4 would imply non-constant curvature. However, I am unsure. Ultimately I would like to assign each of these a quality of a surface that they define such that all surfaces can be "categorized" under some combination of postulates, but that is irrelevant.

I am merely asking:

What two surfaces individually violate the first postulate and violate the fourth postulate?

Prove that if the diagonals of a trapezoid are congruent, then the trapezoid is isosceles, using coordinate geometry.

First, is there a relatively simple formula for the midpoint of two points $a_1$ and $a_2$ in the disk with respect to the hyperbolic geometry? That is, the point on the bisector of $a_1$ and $a_2$ which has minimal distance from $a_1$ and $a_2$ .
Second, if B denotes the degree 2 finite Blaschke product with zeros $a_1$ and $a_2$ , is the critical point of B which is in the disk equal to the midpoint of $a_1$ and $a_2$ ?

How can we use vectors and dot products to show that the diagonals of a parallelogram intersect at ${90}^{\circ <!--\; \circ \; -->}$ if and only if the figure is a rhombus?
I did the proof, but I realized my final answer would be a rectangle. (I know a rhombus is a type of rectangle, too). But I only want to prove the two diagonals are orthogonal.

In the triangle $\mathrm{\angle <!--\; \angle \; -->}A$ is right and D is a point on the side AC such that the segments BD and DC have length equal to 1m. Let F be the point on the side BC so that AF is perpendicular to BC. If the segment FC measures 1m, determine the length of AC.
What´s the length of the segment AC in the triangle below?

In $\mathrm{\Delta <!--\; \Delta \; -->}ABC$, $BE$ and $CF$ are the angular bisectors of $\mathrm{\angle <!--\; \angle \; -->}B$ and $\mathrm{\angle <!--\; \angle \; -->}C$ meeting at $I$. Prove that $AF/FI=AC/CI$
I have tried this question for hours but i am unable to hit the nut
I tried to using: (1)Similarity (2)Angle bisector theorem (3)Relation of lengths of angle bisector :- $B{E}^2+AE.EC=AB.BC$ $C{F}^2+AF.FB=AC.BC$
I tried the question with these many approaches but i unable to prove the above relation.please tell if any of these approach will help or please tell any other method to solve the question.

I was working on a geometry problem relating to the angle bisectors of triangles :

In triangle $\mathrm{\Delta <!--\; \Delta \; -->}ABC$, $\angle A=40°,\angle B=20°,$ and $AB-BC=4$. Find the length of angle bisector from $\angle C$ .

I was able to figure out a majority of the angle measures, but I was unable to utilize the information about the side lengths to find the angle bisector.

Does anyone what method I have to use to solve this?