Questions on the use of algebraic techniques to prove geometric theorems.

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2
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1answer
18 views

Strange “form” of the set of vertices $C(x,y,z)$ such that $ABC$ is a right triangle with hypotenuse $AB$

Let $A(1,-3,4)$ and $B(3,-2,-1)$ and find the set of all $C(x,y,z)$ such that $ABC$ is a right triangle with hypotenuse $AB$ What I did $$AB=(2,1,5)$$ $$BC=(x-3,y+2,z+1)$$ $$AC=(x-1,y+3,z-4)$$ ...
0
votes
0answers
8 views

Analytical geometry, order of topics

I'm planning to write about analytical geometry, but I'm unsure about where to start. The course that I've taken begins with the definition of a vector, it then proceeds to develops all the vector ...
-2
votes
2answers
30 views

Given the endpoints of a line segment, develop the equation of its perpendicular-bisector

Find the equation of the perpendicular bisector of $AB$ for: $A(1\mid 3)$ and $B(-3\mid 5)$. What I did: $m=\frac{3-5}{1+3}=-\frac12$ for the slope of $AB$ $(\frac{3+5}2\mid\frac{1-3}2)=(4\mid -1)$ ...
2
votes
4answers
110 views

Integrating $\sqrt{1-x^2}$ without using trigonometry

I am a beginning calculus student. Tonight I had a thought. Maybe I could calculate $\pi$ using integration, but no trig. The problem is that I don't really know where to start. I thought perhaps I ...
2
votes
2answers
359 views

Algebraic solution to find circle radius given distance of three external points from perimeter

I have an engineering problem, which involves math. The reason it's "engineering" is that I don't need a pure mathematical solution, but a good-enough approximation could work - the only constraint is ...
-1
votes
1answer
29 views

Parabola problem [on hold]

Water squirting out of a horizontal nozzle held $4$ ft above the ground describes a parabolic curve with the vertex at the nozzle. If the stream of water drops $1$ ft in the first $10$ ft of ...
0
votes
1answer
103 views

Lines which intersect the postive half axis of x

We have to find out which lines intersect the positive half axis of $x$. According to this formula we can determine if the angle between two points $(A[x_1, y_1]$ and $B[x_2, y_2]$ ) of the line ...
-1
votes
1answer
1k views

analytic geometry … 2 problems

1st problem : find the equation of the straight line having slope $m$ passing through the point $(a, 0)$. What are the coordinates of the point of intersection of this line with the y-axis? 2nd ...
0
votes
1answer
442 views

Shortest distance between parallel line and plane

I've been doing questions regarding the shortest distance between lines/planes and points , and I've come across a question asking to find the shortest distance between a line and a plane which are ...
2
votes
3answers
92 views
+50

Prove that a point can be found which is at the same distance from each of the four points$\ldots$

Prove that a point can be found which is at the same distance from each of the four points $\bigg(am_1,\dfrac{a}{m_1}\bigg),\bigg(am_2,\dfrac{a}{m_2}\bigg),\bigg(am_3,\dfrac{a}{m_3}\bigg)$ and ...
0
votes
2answers
41 views

Deduce the inequalities $3\lt \pi \lt 12(2-\sqrt{3})$, by calculating the areas of regular twelve-sided polygons.

Calculate the areas of regular dodecagons (twelve-sided polygons) inscribed and circumscribed about a unit circular disk and thereby deduce the inequalities $3\lt \pi \lt 12(2-\sqrt{3})$. This is a ...
6
votes
1answer
51 views

Upper bound on the distance of orthogonal matrices

Dear math stackexchange users, I have a question on orthogonal matrices: suppose I have a matrix $X\in\mathbb{R}^{n\times n}$ and I consider the orbit of the orthogonal group $O(n)$ acting from the ...
0
votes
5answers
160 views

Calculate the angles of a isosceles triangle

In the triangle below, is there a way to calculate the $x$ and $y$? To be more specific, $b = 12.8\rm\,cm\ $ and $h = 10\rm\,cm$, hence $a = 11.87\rm\,cm$. I don't know what to do from here.
3
votes
2answers
220 views

Find $DF$ in a triangle $DEF$

Consider we have a triangle $ABC$ where there are three points $D$, $E$ & $F$ such as point $D$ lies on the segment $AE$, point $E$ lies on $BF$, point $F$ lies on $CD$. We also know that center ...
1
vote
0answers
42 views

A better way to answer this question

So my team and i were asked this question a few years ago on a small Math-A-Thon on my hometown. It went something like this: "We need to transport a neon tube (or any tube, who cares) of 92cm ...
1
vote
2answers
37 views

Given a Line Parametrization, Finding another Equation

So I am given a line $l$ with the parameterization, $x=t, y=2t, z=3t$. Now let some point, $p$ be a plane that contains the line $l$ and the point $(2,2,2)$. So given this, how do I find an equation ...
2
votes
2answers
40 views

Find the Ratio $BM \colon ME$

In Triangle $\Delta ABC$, the Point $D$ is on $BC$ such that $D$ divides $B$ and $C$ in the Ratio $1 \colon 3$ and there is a point $E$ on $CA$ such that $E$ divides $C$ and $A$ in ratio $1 \colon 3$. ...
16
votes
1answer
262 views

With what analytic functions can one construct the $(x,y)$ coordinate axes using a straightedge and a compass?

Given the graph of $y = \frac{1}{x}$, construct the $(x,y)$ coordinate axes using a straight edge and a compass. The solution to this problem is known (mouse over the spoiler text below for a ...
0
votes
1answer
24 views

Equation of the locus

Find the equation of the locus of a point $P = (x, y)$ when the sum of the squares of the distances from $P$ to the points $(a, 0)$ and $(-a, 0)$ is $4b^2$, where $b \geq \dfrac{a}{\sqrt{2}}$?
3
votes
1answer
41 views

How to get rid of the term with $xy$?

I'm trying to put this conic on an identifiable form. $$4x^2-4xy+y^2+20x+40y=0$$ I know that the term $xy$ implies that I need to rotate the conic so that $xy$ vanishes. But I've read on some books ...
0
votes
0answers
20 views

Should the expanded expression of a quadratic form be equals to It's original expression?

Sorry if the question is a little misleading, but I have no better way to express it. The text below should clarify. Suppose I have the equation of a conic: $x^2+y^2+z^2-2x+3y+z+2=0$, with this I ...
4
votes
1answer
34 views

Start and end point of a rotated ellipse

I have the data of an incomplete ellipse and I need to retreive the minimun information in order to describe an elliptical arc. In particular following are my ellipse data: Major axis vector (x, y) ...
0
votes
1answer
8 views

Point within a Cube in 3D environment

I have a cube in 3D space with 8 corner points with their X,Y,Z Coordinates. I know how to test if any given point lies inside a cube by Comparing their coordinates to be greater or smaller than the ...
0
votes
3answers
885 views

Find the area of the triangle using analytic geometry [on hold]

We have a $\triangle ABC$ with: Base $AB$ with length 14 $AC$ with length 15 $BC$ with length 13 Find the area of the triangle using analytic geometry.
2
votes
1answer
15 views

Where are these choices of $A',B',C'$ for this quadratic form?

I'm studying quadratic forms: In the book I'm reading, he starts by looking at quadratic forms such as: $$\varphi (x,y)=Ax^2+2Bxy+Cy^2$$ And that given this quadratic form, one can introduce via ...
3
votes
2answers
20 views

The Reason for different Forms of Equations

I recently started learning about conic sections and saw people writing the equations for the different figures (circle, parabola, ellipse, and hyperbola) in different forms. (standard form, vertex ...
1
vote
1answer
21 views

For line $ax+by+k=0$ which intercepts form a triangle rectangle with area $A$, find $k$

I know that the area of a triangle is given by the formula $A=\frac{1}2Bh$ and the intercepts of line $ax+bx+k=0$ are $(B,0)$ and $(0,h)$ which forms a square with area $2A$, but without brute-forcing ...
0
votes
1answer
36 views

When should I shift $a$ and $b$ in $\cfrac{x^2}{a^2}+\cfrac{y^2}{b^2}=1$?

Find the reduced equation of the elypsis such that: The foci are $(0,6);(0,-6)$ and the larger axis has length $34$. I did the following: Taking the equation ...
1
vote
0answers
27 views

Question about the coordinates in a new origin on the plane.

I'm reading a book on analytic geometry, specifically on a chapter on change of coordinates. It says that having the origin $O$, one point $P$ and a new origin $O'$, the vector that describes the ...
1
vote
1answer
23 views

A simplified formula for area of triangle when equations of the sides are given

For i = 1, 2, and 3, let $a_ix + b_iy + c_i = 0$ be three equations of 3 (non-special cased) straight lines. From which, the co-ordinates of the vertices can be found. Using these co-ordinates, via ...
2
votes
1answer
54 views

What geometric object is given by this equation?

What geometric object is given by this equation? $x^2+y^2+z^2+2xy+2xz+2yz-x-y-z-6=0$ Maple says it's a hyperboloid of one sheet, but is there a way to show it without going the long way by using the ...
1
vote
1answer
28 views

Why does this hyperboloid change into a surface? [duplicate]

Given this equation $x^2+y^2+z^2+2xy+2xz+2yz-x-y-z=6$ and the corresponding quadric: If I rearrange the equation to $(x+y+z-3)(x+y+z+2)=0$ (which is equivalent), I get: So, which is the right ...
2
votes
2answers
28 views

Find the equation of base of Isosceles Traingle

Given the two Legs $AB$ and $AC$ of an Isosceles Traingle as $7x-y=3$ and $x-y+3=0$ Respectively. if area of $\Delta ABC$ is $5$ Square units, Find the Equation of the base $BC$ My Try: The ...
1
vote
1answer
19 views

Points on two skew lines closest to one another

Given two skew lines defined by 2 points lying on them as $(\vec{x}_1,\vec{x}_2)$ and $(\vec{x}_3,\vec{x}_4)$. What are the vectors for the two points on the corrwsponding lines, distance between ...
3
votes
0answers
39 views

How to transform (rotate) this hyperbola?

Given this hyperbola $x_1^2-x_2^2=1$, how do I transform it into $y_1y_2=1$? When I factor the first equation I get $(x_1+x_2)(x_1-x_2)=1$, so I thought $y_1=(x_1+x_2)$ and $y_2=(x_1-x_2)$. ...
-1
votes
2answers
29 views

Co-ordinate geometry and area of triangle

When a straight line $ax+by+c=0$ forms a triangle with the axes $x$ and $y$, what is the general formula for the area of the triangle?
0
votes
1answer
26 views

Area of Triangle Given 3 vertices

Given that $P=(1,1,0), Q=(1,0,1), R=(0,1,1)$. I need to find the area of the triangle. What I have done: I have tried finding the distances of PQ, QR, and PR. I have those distances, I don't know ...
0
votes
0answers
35 views

Error Distribution of Canny's algorithm in some borders?

Assume you have two circles which are filled with many ellipses of non-arbitrary size from a finite set. How can you deduce the distribution of the difference of circles' diameters/areas in theory? ...
3
votes
2answers
81 views

Let $ S=\{(x,y)\in\mathbb{R}^2 \ | \ x^2+y^2=1 \text{ and } y\geq 0\}$. Determine $S+S+…+S $.

Let $$ S=\{(x,y)\in\mathbb{R}^2 \ | \ x^2+y^2=1 \text{ and } y\geq 0\}$$ By the usual notation for sum of sets let $$ 2S\overset{\text{not}}{=}S+S=\{(x_1+x_2,y_1+y_2) \ | \ (x_1,y_1), ...
-2
votes
0answers
34 views

Is the focal distance of $x^2+\cfrac{2y^2}{3}=8$ equal to $2\sqrt{-4}$?

I've just computed the focal distance of $x^2+\cfrac{2y^2}{3}=8$ and found $2\sqrt{-4}=4i$ as follows: $$x^2+\cfrac{2y^2}{3}=8\\ \cfrac{x^2}{8}+\cfrac{y^2}{12}=1$$ Then the focal distance should be ...
10
votes
3answers
5k views

The shortest distance between any two distinct points is the line segment joining them.How can I see why this is true?

On a euclidean plane, the shortest distance between any two distinct points is the line segment joining them. How can I see why this is true?
6
votes
0answers
51 views

Is this solution legal?

Let $M(1,-1)$ be a point in a plane. Find its distance from a line given by $x+2y-4=0$. Later on I found a formula: $$d=\frac{\left | Ax_{0}+Bx_{0}+C \right | }{\sqrt{A^2+B^2}}$$ But I did it ...
0
votes
0answers
27 views

Algorithm to calculate line segments between two points bounded by multiple surfaces

Problem statement: As a specific case, let's say I have a volume composed of a series of concentric cylinders. Given a fixed point P (a,b,c), and another randomly sampled point Q (x0,y0,z0), I would ...
0
votes
0answers
33 views

Proof regarding hyperbolas

Given the parameters $a,b>0$ we set $c:=\sqrt{a^2+b^2}$ and $e:=\large\frac{c}{a}$ (eccentricity), the focal points are $F=(c,0)$ and $F'=(-c,0)$, the directrix $L$ with the equation ...
1
vote
0answers
18 views

Why $\|X-F\|=e|(X-F)\cdot N -d|$ should be written as $\|X-F\|=e|(X-F)\cdot N +d|$?

I'm reading Apostol's Calculus. $\quad $ And I've tried to do the following exercise: $\quad \quad \quad \quad $ I am a little confused: I have the portuguese version of the book, and it ...
1
vote
3answers
34 views

Finding extrema.

Find the minimum distance between point $M(0,-2)$ and points $(x,y)$ such that: $y=\frac{16}{\sqrt{3}\,x^{3}}-2$ for $x>0$ . I used the formula for distance between two points in a plane to get: ...
0
votes
0answers
10 views

distance between planes in a simplex

In an Euclidean space there are n points at equal distances d to each other (regular simplex). Find out a distance between two parallel planes, one spanned at points numbered 1 through k, the other at ...
0
votes
0answers
12 views

4-dimensional simplex

In a 4-dimensional Euclidean space, there is a simplex, with given lengths of all the edges aij = distance(Ai,Aj). Find a distance between gravity centers of sides, opposite to each other. Notice: ...
2
votes
1answer
113 views

Finding the location of the end of an arc, knowing the beginning, the arc's length and the radius

I apologise in advance if this is really basic. I have a circle of radius $15$, from which I work out an arc, given an angle of arbitrary value (it's for a computer program). Given that I know the ...
0
votes
0answers
21 views

Help with Apostol's “Calculus, vol. 1”, Section 1.18

In section 1.18 ("The area of an ordinate set expressed as an integral"), Apostol proves two theorems. the first, theorem 1.10, deals with the area of a function's ordinate set; the second, theorem ...