For questions about geometric shapes, congruences, similarities, transformations, as well as the properties of classes of figures, points, lines, angles.

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5 views

How can I use menelaus' theorem here?

Given 4 points on a circle A, B, C, and P. Draw the orthogonal projections of P onto triangle ABC and call them $P_1, P_2,P_3$. Show that $P_1, P_2,P_3$ are collinear. After drawing this out, I ...
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1answer
8 views

Proving that $2$-D parabolic coordinates are orthogonal

How can we prove that the parabolic coordinate system in two dimensions is orthogonal? I tried using the dot product, but don't know where to start or what basis vectors can be used in two dimensions. ...
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1answer
28 views

How would I use vectors for this geometry problem?

Consider a quadrilateral ABCD. K, L, M, N are the midpoints of the segments AB, BC, CD, DA respectively. O is the intersection point of LN, KM. Let P and Q be the middle points of the diagonals AC ...
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2answers
32 views

Trisecting the sides of a triangle.

Consider the hexagon formed by the six points which trisect the sides of a triangle(two on each side). Is is true that when we connect opposite points in this hexagon, the lines intersect at a single ...
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2answers
32 views

Distance between four points

I have four points as shown in this figure: I want to calculate one vector for all these points. So, what would be the correct way: 1) I take the vector between $A-B, B-C, C-D$ and add them $(A-B ...
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0answers
20 views

Is there a name for this special, “most parallel” ultraparallel line in hyperbolic geometry?

Suppose you're in the hyperbolic plane, and you have a line L and a point P not through L. There are an infinite number of lines parallel to L that go through P. However, there's one line M which is ...
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4answers
58 views

Is it possible to find the area of a shape from its perimeter?

Is it possible to find the area of a free form shape knowing the perimeter? An example would be a clover leaf shape. If the perimeter is 96 how would I know what the area would be?
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0answers
18 views

a valuable question about differential geometry, the curve$\beta(s)=\alpha(s)-rn(s)$

Let $\alpha(s)$,s$\in [0,l]$ be a closed convex plane curve positively oriented. The curve $$\beta(s)=\alpha(s)-rn(s)$$,where r is a positive constant and n is the normal vector, is called a parallel ...
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0answers
25 views

27 Lines on a Cubic

In Ravi Vakil's notes, there is a proof (in section 27, of course) of the famous result that every nonsingular cubic hypersurface in $\mathbb{P}^3_k$ over an algebraically closed field $k$ has exactly ...
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0answers
14 views

Intercept planet following an elliptical path (i.e. interplanetary space travel)

So, just as in this question (Intercept path to object following an elliptical path) I have a simple game where I want spaceships to intercept planets, which follow elliptical paths (in my case ...
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3answers
33 views

The lines with equations $y = 5x − 6$ and $10x + cy = 8$ are perpendicular, find c [on hold]

The lines with equations $y = 5x − 6$ and $10x + cy = 8$ are perpendicular. Find the value of c. Well, I am not sure even where to start
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0answers
37 views

How can I find the common axis of 2 cones in space that have the same base radius but different heights?

How do I find the 3D vector describing the axis of 2 overlapping cones, like this: If I have only the following information: Coordinates of the common tip Coordinates of a point on the yellow ...
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0answers
10 views

Showing there is a cartesian coordinate system on EG.

I'm pretty sure I should just show there is a bijection between the points in EG and elements of R^2. How do I do this? note: EG=Euclidean Geometry
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0answers
17 views

Questions on curvilinear asymptotes

I just saw curvilinear asymptote which sort of fascinated me. A little bit of thinking raised two questions for which I couldn't get the the answer by googling. Is there a general method to find a ...
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1answer
21 views

Line between points in projective space?

I am trying to find the line through the points $(0 : 1 : 0)$ and $(1 : 1 : 1)$ in $\mathbb P^2$ and $(0 :1 : 0: 1)$ and $(1: 1: 1: 0)$ in $\mathbb P^3.$ Would the first line be the set of points ...
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0answers
24 views

Formula to calculate angle on a fan or semicircle

How do I calculate the angle shown in the picture given the height, width, and the arc deduction of $2$? I had applied the Right Triangles formula to calculate the hypotenuse: $h^2 = a^2 + ...
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1answer
11 views

find a point such that the distance between this point and a plane is equal to the distance between another point and a line [on hold]

I have: $r:(2,1,0)+\lambda(0,4,-3) \qquad \pi:4y-3z-4=0 \qquad A=(2,4,4)$. I have to find a point $C$ such that $d(C,\pi)=d(A,r)$ where $d$ is the distance
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0answers
6 views

Can Magnification/Scaling (transformation) be prepresented by a vector?

Vectors represent three bits of information: Magnitude, Line of Action, and Direction. A Translation (transformation) can be represented by a vector: object is moved By so much (magnitude) Along a ...
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0answers
13 views

Parametric equations for hypocycloid and epicycloid

Suppose that the small circle rolls inside the larger circle and that the point $P$ we follow lies on the circumference of the small circle. If the initial configuration is such that $P$ is at ...
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0answers
11 views

For each point on a line there exists a unique perpendicular line through that point

I'm trying to show that in an absolute plane (only the first four axioms without the parallel axiom hold) for each Point $P\in l$ there exists exactly one perpendicular line through $P$. My idea was ...
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1answer
11 views

Given a number of vertices , a radius, and rotation calculate vertices' coordinates for regular polygons

So, I know half the answer to this but I don't know how to adjust it for rotation. I believe formula the below is correct if I did not have to take into account rotation. $r \cos(2 \pi i / n) = y$ ...
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0answers
13 views

Point on ellipse after walking a distance on the perimeter [duplicate]

I've the equation of an ellipse. Given a point (x,y) on the ellipse and a length L , I want to find the coordinates (x1,y1) of the point where I'd end up after taking a walk of length L from (x,y), ...
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0answers
17 views

Find Formal Proof on Loci Theorems

Please help me to prove this theorems of loci Theorem 12 The locus of a point at a given distance from a given line is two lines parallel to the given line and at the given distance from it. Theorem ...
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0answers
7 views

$\frac{4}{3}(1,-1,0)^T+\textrm{vect} \left((-1,1,-1)^T\right)$

Here I have a passage on the conclusion, but I do not understand it. the conclusion say : $f$ is the oriented axis of screw.. how I can from $\frac{4}{3} \begin{pmatrix}1\\1\\0 ...
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0answers
48 views

Can ${n \choose 2}$ points be covered by lines determined by $n$ points?

Let $S=\{(a_1,b_1),\ldots,(a_{n \choose 2},b_{n \choose 2})\}$ be ${n \choose 2}$ points on the plane. Does there exist $n$ points, such that the lines determined by the $n$ points cover all the ...
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0answers
8 views

union of two Geodesics

Let $X$ be a metric space. $a,b,c \in X$ and $\sigma_1 , \sigma_2$ geodesics from $a$ to $b$. Let $\sigma$ geodesic from $b$ to $c$. Want to show : $\sigma_1 \cup \sigma $ geodesic $\Rightarrow ...
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1answer
17 views

Length of tendon in circle [on hold]

What is the length of chord that pass on two specific point. For example I have circle ( r=1) point1 :(x1,y1) point2(x2,y2); length of chord?
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0answers
20 views

a question about differential geometry, the relation between osculating plane and the points of $\alpha(s)$

question:please prove the limit position of the circle passing through $\alpha(s)$,$\alpha(s+h_{1})$,$\alpha(s+h_{2})$ when $h_{1}$ and $h_{2}$ approaches 0 is a circle in the osculating plane at s, ...
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0answers
13 views

Is kahler geometry related to finsler geometry? [on hold]

Is kahler geometry related to finsler geometry? Is the metric $\partial_a\phi\partial_b\phi$ always kahler?
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2answers
44 views

If a curve $\gamma$ through two points $P,Q$ satisfy $\|Q-P\| = \int^{t_1}_{t_0} \| \gamma^{'} \| \, \text {d}t$, then $\gamma$ is a straight line?

In a theorem called "A straight line is the shortest curve through two given points", I prove that for any two points $P,Q \in \mathbb R^2$ and any curve $\gamma : (a,b) \rightarrow \mathbb R^2$ with ...
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1answer
41 views

Average distance to a random point in a rectangle from an arbitrary point

I'm interested in the mean distance between an arbitrary 2D point, $(p, q)$, and a uniformly distributed point inside a rectangle defined by the lower left and upper right vertices $(x_0, y_0)$ and ...
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1answer
32 views

Trying to define a simple “warp” function

I'm trying to define a 2D "warp" function y=f(x,w). A picture is worth a thousand bytes: I am looking for a simple function f(x, w) that satisfies the ...
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2answers
14 views

Geometry: What is the height of a solid pyramid

So the question is OPQRS is a right pyramid whose base is a square of sides 12 cm each. Given that the the slant height of the pyramid is 15cm. And now I need to find the Height of the pyramid. I ...
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0answers
60 views

Prove that.. Please quick. [on hold]

If the length of perpendicular from the point $(1,1)$ to the line $ax-by+c=0$ be $1$, show that $1/c+1/a-1/b=c/2ab$. if $p$ and $p'$ be the length of the perpendiculars from the origin upon the ...
2
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0answers
27 views

Unique perpendicular line

Consider an absolute plane (i.e. it satisfies all axioms except the parallel axiom). Let g be a straight line and P a point not in g. Then there is a unique straight line going through P which is ...
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1answer
13 views

Inscribed Rhombus

Can a rhombus that is NOT a square be inscribed in a circle? A quadrilaterals opposite angles must add up to 180 in order to be inscribed in a circle, but a rhombuses opposite angles are equal and do ...
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1answer
23 views

Prove that there is exactly one perpendicular line

Consider an absolute plane (i.e. it satisfies all axioms except the parallel axiom). Let $g$ be a straight line and $P$ a point not in $g$. Then there is a unique straight line going through $P$ which ...
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0answers
6 views

Calculating the position of the ascending and descending nodes in an orbit

I'm working on a space sim and I'm stuck getting the position of the Ascending and Descending orbital nodes. I know how to get the position at any angle of the orbit, but it's measured from the ...
2
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1answer
18 views

Proof: number of intersection points of borders of a convex polygon and its translate never greater than 2

How can I prove the observation that the borders of a convex polygon $P$ with no parallel sides in $\mathbb{R}^2$ and a translate of $P$ by any vector $t\neq 0 \in \mathbb{R}^2$ that is not parallel ...
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2answers
31 views

When the intersection between a sphere and a cylinder is planar?

We have a sphere and a circular cylinder. Let the sphere center be $O$ and radius $R$, and the cylinder axis $a$ and radius $r$. I solved the specific case intersection graphically on 2 planar ...
2
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1answer
62 views

Calculationg the angle of a triangle

I am trying to find a specified angle of a triangle. In triangle $ABC$, $\angle A = 20^\circ$. $D$ and $E$ are points on $AB$ and $AC$, where $AB=AC$. $\angle EBC = 50^\circ$ and $\angle DCB = ...
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1answer
14 views

Calculate perimeter of rhomboid

I am trying to solve the following problem but I got stuck In a rhomboid with an area of $48 \space cm^2$, the major diagonal is $4$ cm shorter than the double of the minor diagonal. Calculate the ...
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2answers
40 views

How do I solve the triangle?

Let $ABC$ be a given triangle, such that $AD=3$, $DE=5$, $EC=24$ and $∠ABE=90^\circ$, $∠DBC=90^\circ$, where $D$ and $E$ are points on $AC$ (and$ D$ is between $A$ and $E$). Then, find the length of ...
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2answers
49 views

Counter example to Mostow's rigidity theorem for 2-manifolds.

I am trying to understand a counter-example to Mostow's rigidity theorem. Here is the counter example I want to understand. Take two non-isometric octagons with the sum of interior angles equal to ...
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0answers
19 views

Get a function (equation) from data points?

Is there a way to get a function (equation) from data points? For example, we have this famous Google's 'Batman' function: ...
2
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0answers
36 views

Rational functions over curves define regular maps?

I've a rather simple question that is strucking me, perhaps because I'm a newbie in algebraic geometry. If we have a projective irreducible non-singular curve $C\subseteq \mathbf{P}^n_k$ where $k$ is ...
3
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1answer
31 views

Origin of period function model of primes

There is a web page attributed to Omar Pol, "Sobre el patrón de los números primos: Determinación geométrica de los números primos y perfectos." ("On the pattern of primes: Geometric Determination of ...
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0answers
15 views

What's another name for the non-negative quadrant, or any non-negative orthant?

Is there another canonical name for the non-negative quadrant, or for the generic non-negative orthant?
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0answers
19 views

Calculate change in distance to camera from scale

I have 2 pictues of an object. In one picture the object is close and in the other one it is far away. I have these values: $z$ = distance of object to camera in first picture $f$ = focal length ...
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0answers
52 views

one question about differential geometry,show the curvature k($\phi$)

One often gives a plane curve in polar coordinates by $p=p(\phi)$,$a\le\phi \le b$. (1)Show that the arc length is $$\int_{a}^{b}\sqrt{p^2+\dot p^2}$$,where $\dot p$ means the derivative of p with ...