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

learn more… | top users | synonyms (1)

6
votes
1answer
384 views

Why is $m$ used to denote slope?

What is the reason, historically, that the letter $m$ is used to denote the slope of a line?
2
votes
0answers
45 views

degeneracy loci of dimension $2$

Let $X$ be a smooth complex projective variety of dimension $n \ge 4$ and let $F$ and $E$ be two (holomorphic) vector bundles of rank $f$ and $e$ over $X$. Given a morphism $\varphi: F \to E$ of ...
3
votes
1answer
1k views

Use Pappus' theorem to find the moment of a region limited by a semi-circunference.

This is part of self-study; I found this question in the book "The Calculus with Analytic Geometry" (Leithold). $R$ is the region limited by the semi-circumference $\sqrt{r^2 - x^2}$ and the ...
3
votes
1answer
493 views

Maximum cosine for angle between 2 vectors when 1 vector is partially unknown

assuming I have two vectors $A$ and $B$, where $A$ is completely known and from $B$ I know only that the first k components are 0. What is the maximum possible cosine value for the angle between the ...
5
votes
4answers
756 views

Why do all circles passing through $a$ and $1/\bar{a}$ meet $|z|=1$ are right angles?

In the complex plane, I write the equation for a circle centered at $z$ by $|z-x|=r$, so $(z-x)(\bar{z}-\bar{x})=r^2$. I suppose that both $a$ and $1/\bar{a}$ lie on this circle, so I get the equation ...
7
votes
3answers
1k views

How does this equality on vertices in the complex plane imply they are vertices of an equilateral triangle?

I've read that if the complex numbers $a_1$, $a_2$ and $a_3$ are the vertices of a triangle in the complex plane such that $$ a_1^2+a_2^2+a_3^2=a_1a_2+a_2a_3+a_1a_3 $$ then the vertices are actually ...
1
vote
0answers
75 views

Translate group definition into geometry system

I need to reword the definition of group (the four axioms: closure, associativity, identity and invertibility) to be lines and points of non-Euclidean geometry (the axiom system defined as geometry). ...
3
votes
2answers
221 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 ...
2
votes
4answers
8k views

How to Determine an Equation of a Circle using a Line and Two Points on a Circle

My question goes like this: Determine the equation of a circle tangent to the $x$-axis and passing through $(5,1)$ and $(12,8)$. I need not only the answers, but also the steps on how you did it so ...
0
votes
1answer
167 views

Vectors problem

can anyone help me with this problem: Is it possible to construct three vectors (a,b,c) in 3D, such that angle between a and b is 30 degrees, between a and c is 150 degrees, and between b and c is 30 ...
1
vote
1answer
118 views

Circle locus, how to satisfy the equation.

$A(-3,1), B(0,-5), P(X,Y)$ If $|AP| = 2|BP|$ prove that $x$ and $y$ satisfy the equation: \begin{aligned} \ x^2+y^2-2x+14y+30 =0 \end{aligned} I get as far as determining the ...
1
vote
2answers
1k views

Finding & Plotting equation of hyperbola given foci, and difference in distances between them.

I have to plot the hyperbola (3 of them actually) in MATLAB, and so it'd be good if I could find some sort of general formula. The foci do not necessarily have to be on the axes (e.g. $(5,3)$ and ...
1
vote
2answers
596 views

Stereographic projection of a regular tetrahedron inscribed in the Riemann sphere?

I've been reading about stereographic projections. I did a problem about finding the stereographic projection of a cube inscribed inside the Riemann sphere with edges parallel to the coordinate axes. ...
1
vote
1answer
272 views

How to scale a polyhedron contained a 3-sphere?

In the 3-sphere simulator I am building, the viewpoint is contained in the space of a 3-sphere (the surface of a 4-D hypersphere), and the user is able to navigate through it. There are some ...
3
votes
1answer
1k views

Equation of a sphere as the determinant of its variables and sampled points

Searching for an equation to find the center of a sphere given 4 points, one finds that taking the determinant of the four (non-coplanar) points together with the variables $x$, $y$, and $z$ arranged ...
3
votes
1answer
178 views

Algebra question about Triangle Interiors

I was reading about Triangle Interiors on Wolfram Alpha: http://mathworld.wolfram.com/TriangleInterior.html and they have a simple equation: $$\mathbf{v} = \mathbf{v}_0 + a\mathbf{v}_1 + ...
0
votes
2answers
797 views

Given the cartesian coordinates of four points, how to calculate the interection of two lines they form?

Given four complex numbers $A, B, C, D$ interpreted as points on the plane, how can I calculate the number that represents the intersection of the lines formed by $A, B$ and $C, D$?
0
votes
2answers
139 views

is this equation solvable?

Can someone please solve these 2 equations to get values of h and k? I know the values of h and k but not sure how to solved these equations to get h and k 's values $(20.01 - h)^2 + (17.94 - k)^2 = ...
1
vote
3answers
115 views

calculating a point on circumference

See the diagram Known values are A: (-87.91, 41.98) B: (-104.67, 39.85) C: (-96.29, 40.92) L: 14.63 // L is OC Known angles ...
2
votes
5answers
2k views

finding center of circle

How can I calculate center of a circle $x,y$? I have 2 points on the circumference of the circle and the angle between them. The 2 points on the circle are $P_1(x_1,y_1)$ and $P_2(x_2,y_2)$. The ...
3
votes
2answers
2k views

Altitudes of a triangle are concurrent (using co-ordinate geometry)

I need to prove that the altitudes of a triangle intersect at a given point using co-ordinate geometry. I am thinking of assuming that point to be $(x,y)$ and then using slope equations to prove ...
0
votes
2answers
230 views

Vector Geometry - relation between a point and a line with angle and one known point on it

I have two problems I will be very grateful if somebody helps me about them. If I have a line $L_1$ with a known point $(x_1, y_1)$ on it and has slope $\theta_1$, how do I know if a point $P=(x, y)$ ...
2
votes
2answers
214 views

Rigorously showing there are infinitely many points of intersection?

I'm working on a problem that states if $k\geq 3$, $x,y\in\mathbb{R}^k$, $|x-y|=d>0$, and $r>0$, then (a) If $2r>d$, there are infinitely many $z\in\mathbb{R}^k$ such that ...
2
votes
2answers
278 views

Conditions for intersection of parabolas?

What are the conditions for the existence of real solutions for the following equations: $$\begin{align} x^2&=a\cdot y+b\\ y^2&=c\cdot x+d\end{align}$$ where $a,b,c,d $ are real numbers. ...
3
votes
2answers
11k views

How do I find the equation of a tangent line to a curve?

I'm given $x^2+2x-4$ at $x=2$ and I have to find the tangent line to this curve at that point...
0
votes
1answer
134 views

lines intersection

I have to find intersection of two lines ($AH$ & $CD$) $A(3.42,-1.84,8.56) $ $B(-3.42,3.84,-8.56) $ $C(0.00,16.25,0.00)$ $AH$ is the perpendicular; $CD$ is the median I tried so: Firstly I ...
-1
votes
1answer
150 views

Formula to show square root between 2 values

Bear with me - it's been a while since I did this at school! I need to plot a curve in the form of a square root (kind of an 'r' shape if you will) I have 6 intervals along my x axis, and my maximum ...
1
vote
1answer
113 views

Mirorring a set of Points

Let's say I have a cloud of points, and I know the equation of the symmetry plane. I'd like to mirror every single point with respect to this plane. It might be much simpler than I think, but I have ...
2
votes
1answer
412 views

Need help with the proof of conic section

Prove that the intersection of a plane and a object consist of one cone and one upside-down cone where the tip of cone meet is either degenerate conic or conic Also, idenify in what situation, the ...
2
votes
1answer
201 views

What is wrong with this proof that isometries must be surjective?

Let $\phi : \mathbb{R}^2\rightarrow\mathbb{R}^2$ be an isometry. Suppose $\phi$ is not surjective, that is there exists some $v \in \mathbb{R}^2$ whose fiber $\phi^{-1}(v)$ is empty. Then by the ...
0
votes
2answers
148 views

How to see a plane is tangent to a sphere from their equations

Say you have two equations with three variables, the first is the equation of the surface of a sphere and the second of a plane. In this case they intersect in a point $(1,0,0)$. The only way I know ...
2
votes
0answers
500 views

projection of a sphere onto a plane

Consider you have a sphere centered at the origin.The sphere has a diameter of $\frac{1}{2} \sqrt{\frac{3}{2}}$. This means that the inscribed cube has an edge of 1. Take any point from the plane ...
5
votes
5answers
1k views

Distance Between A Point And A Line

Any Hint on proving that the distance between the point $(x_{1},y_{1})$ and the line $Ax + By + C = 0$ is , $$\text{Distance} = \frac{\left | Ax_{1} + By_{1} + C\right |}{\sqrt{A^2 + B^2} }$$ What ...
4
votes
3answers
236 views

Points at integer distance

How many points can one can place in $\mathbb{R}^n$, with the requirement that no $n+1$ points lie in the same $\mathbb{R}^{n-1}$-plane, and the euclidean distance between every two points is an ...
0
votes
2answers
105 views

Calculate Points for a Parallel Line

Given a line running through p1:(x1,y1) and p2:(x2,y2), I need to calculate two points such that a new parallel line 20 pixels away from the given line runs through the two new points. Edit: The ...
0
votes
1answer
1k views

Find point on sphere with directional tangent vector

Say a sphere equation like this: $x^2+y^2+z^2=5$. I want to find a point on the sphere whose tangent vector is perpendicular to the vector $\begin{bmatrix} 2\\ 3\\ 4 \end{bmatrix}$. I go ...
3
votes
1answer
56 views

Segments on a plane, what curve do the intersections tend to?

In a Cartesian diagram, given a size $s$, suppose I create $m$ segments as such: I connect $(0,s/m)$ with $(s,0)$; $(0,2s/m)$ with $(s-s/m,0)$; ... ; $(0,s)$ with $(s/m,0)$. For example, if $s=4$ ...
1
vote
0answers
548 views

Calculation of the coordinates on the surface of a tilted cone

I have a mathematical problem (which I am trying to solve with Mathematica). I want to tilt a cone around its base point as in my example, where I have used Mathematica's Cone-function and spherical ...
0
votes
1answer
19 views

What is $a$ in the formula for the distance of point to plane formula: $h=\frac{|(a-p) \cdot n|}{|n|}$?

What does $a$ stand for in the following formula for the distance of a point to a plane? $$h = |PF| = \frac{|d - p \cdot n|}{|n|} = \frac{|(a-p)\cdot n|}{|n|} .$$
0
votes
2answers
209 views

Hyperbola property

I am posting the following question under homework category. I hope I will have very good answer from mathematicians about conic sections. I have seen closely the conic sections and their ...
1
vote
1answer
226 views

This is the most difficult question I could get without using mass point geometry

In triangle ABC, points D and E are on sides BC and CA respectively, and points F and G are on side AB with G between F and B. BE intersects CF at point O_1 and BE intersects DG at point O_2. If FG ...
3
votes
3answers
310 views

Apostol Section 13.25 #13 - Conic Sections

Question: Prove that a similarity transformation (replacing $x$ by $tx$ and $y$ by $ty$) carries an ellipse with center at the origin into another ellipse with the same eccentricity. (The next ...
-2
votes
1answer
1k views

How to prove we could use mass point geometry to solve all the triangle problem involving ratio between line segment and transversal in a triangle?

what is an easy way to prove that use mass point geometry to solve a problem in the link i provide that is involving cevians in a triangle is same as using the other way in euclidean geometry or ...
1
vote
2answers
39 views

Curve equation - help in understanding

OK, I wasn't on a class regarding this type of excercises. I got the notes from the lesson but have no idea how is it working. I hope you'll be able to clarify: Determine the equation of the curve ...
1
vote
2answers
288 views

Is there a generalized method of rotation for curves?

I know that we can rotate a curve in $R^2$ about a linear axis, as is common for first year calculus problems involving solids of revolution. But has anyone come up with a general method to take a ...
1
vote
0answers
384 views

Solving a system of quadratic equations

I'm facing a rather trivial problem which I seem unable to solve... Not being a mathematician (but an engineer with a bit of knack for math), I managed to formulate it in a way that seemed solvable to ...
1
vote
1answer
65 views

If $\|\mathbf{OA}+k\mathbf{OB} \|=1$, prove that $\text{Area}(OACB) \leq \| \mathbf{OB} \|$

OACB is a parallelogram. In other words if $\left \|\mathbf{a}+k\mathbf{b} \right \|=1$ ($k\in\mathbb{R}$), prove that $$\|\mathbf{a}\| \cdot \|\mathbf{b} \| \cdot \sin \theta \leq \|\mathbf{b} \| $$ ...
0
votes
1answer
493 views

How to solve such an equation ? (Line-Plane Intersection)

I don't know how to solve such an equation: $$ t = - \frac{ ...
5
votes
6answers
1k views

Why, conceptually, do limaçons $r=a+b\cos\theta$ have dimples when $|\frac{a}{b}|<2$?

Using calculus, I can justify that limaçons—the polar graphs of $r=a+b\cos\theta$ for various nonzero real values of $a$ and $b$—are dimpled when $|\frac{a}{b}|<2$, but that doesn't seem to yield ...
4
votes
2answers
359 views

Why do definitions of distinct conic sections produce a single equation?

I understand how to get from the definitions of a hyperbola — as the set of all points on a plane such that the absolute value of the difference between the distances to two foci at $(-c,0)$ and ...