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

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2
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2answers
31 views

Using the cross product to find the angle between two vectors in $\Bbb R^3$

Let $$u = \langle 1, −2, 3 \rangle \qquad \text{and} \qquad v = \langle −4, 5, 6 \rangle$$. Find the angle between $u$ and $v$, first by using the dot product and then using the cross product. ...
0
votes
0answers
20 views

Prove winding number is the same as index of a vector field.

I'm trying to prove that the winding number and the index around a point in a vector field are the same. I know that the index is sometimes defined as the winding number but I'm working with the ...
2
votes
2answers
31 views

Parallelogram ABCD

There's a parallelogram $ABCD$. I'm given point $A(3,12)$ and point $B(-1,5)$. Given the equations of the lines $BC$ and $AC$ are $y=8x+13$ and $y=3x+3$ respectively. How to find the coordinates of ...
0
votes
1answer
11 views

Show that tetrahedral has a segment perpendicular to a plane

In this tetrahedral, I have that $$DC = DA, AB = BC$$ and also, I have that angle $DBA$ is $90^\circ$. I need to show that at least one segment is perpendicular to a plane in this tetrahedral. ...
0
votes
3answers
23 views

Derivation of the equation for the envelope

Suppose we have a family of curves on the plane. The equation of the curves is given by $$ f(x ,y ;t) = 0 . $$ Here $t$ is the parameter. On Wiki, the equations determining the envelope of this ...
1
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2answers
54 views

Area of regular n-gon without trig?

As the title suggests I'm trying to find a formula for the area of a regular n-gon that doesn't use trigonometry. I already know the trig formula and I realize that my question is simply asking for ...
2
votes
1answer
68 views

How can I find $\det(A)/\det(B)$, when individual determinants blow up

I am interested in the quantity: $\frac{\det(A)}{\det(B)}$ of positive definite matrices $A$ and $B$. The problem I am running into now is that for large $A$,$B$, (around $200 \times 200$), the ...
3
votes
1answer
48 views

Let $S$ be a set of $n$ points in the plane with min spacing of 1. Prove $S$ has a subset of $\ge n/7$ points with min spacing of $\sqrt{3}$.

I believe I have proven the case $n=8,|T|=2$, but welcome feedback. I need help proving the case for general $|T|>2$. From the 2003 Canada National Olympiad: Let $S$ be a set of $n$ points in ...
-1
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1answer
24 views

orthocentre and triangle related question

$AD$, $BE$, and $CF$ are the altitudes of triangle $ABC$ with orthocentre $H$, then $C$ is the orthocentre of which triangle? Answer: triangle $ABH$. Please explain.
1
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2answers
36 views

Given 4 points with 2 on different radius. Obtain the center of the circle.

I'm struggle on a math question that states the following: Black holes have an overwhelming gravity, such that the nearest stars begin spinning around them (Example). Every affected star keeps ...
1
vote
1answer
23 views

What is the name for the image form you get you take a line segment and sweep it through a region of space?

For instance, if you were to take a line segment and translate it along a coplanar path, then you'd get a plane. If the path is cyclic and on that path you rotate the line segment on the axis ...
1
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2answers
45 views

Finding a triangle ABC if $2\prod (\cos \angle A+1)=\sum \cos(\angle A-\angle B)+\sum \cos \angle A+2$

Find $\triangle ABC$ if $\angle B=2\angle C$ and $$2(\cos\angle A+1)(\cos\angle B+1)(\cos\angle C+1)=\cos(\angle A-\angle B)+\cos(\angle B-\angle C)+\cos(\angle C-\angle A)+\cos\angle A+\cos\angle ...
0
votes
2answers
33 views

Generalization of Cantor Pairing function to triples and n-tuples

Is there a generalization for the Cantor Pairing function to (ordered) triples and ultimately to (ordered) n-tuples? It's however important that the there exists an inverse function: computing z from ...
1
vote
1answer
44 views

Stadium Seating - Geometric Sequences

A circular stadium consists of sections as illustrated, with aisles in between. The diagram show the tiers of concrete steps for the final section, Section K. Seats are to be place along every step, ...
3
votes
3answers
61 views

Does the centroid of a triangle ever fall outside of its Morley's triangle?

Let $T$ be a triangle, and $M$ its (first) Morley triangle:                     (Image from Bruce Shawyer web page.) Q1. Does the ...
1
vote
2answers
26 views

Having 2 independent segments made by 4 cartesian points, calculating x points of a smooth curve connecting the two segments

Drawing with an example of what Im trying to do I'm trying to make a sort of turtle program as a toy programming project. I can send instruction to go from A to B straight giving direction and ...
0
votes
0answers
29 views

Calculating amount of cubes that fit in a sphere

I know that the problem of finding out how many spheres can fit in a cube is a commly asked and well documentted ons, but I am struggling to find anything on the inverse of the problem, namely: How ...
2
votes
4answers
799 views

Two circles inside a semi-circle

Two circles of radius 8 are placed inside a semi-circle of radius 25.The two circles are each tangent to the diameter and to the semi-circle.If the distance between the centers of the two circles is ...
0
votes
0answers
18 views

Need help with a design calculations equal spacing of circles

I need the spacing between the circles to match. Design need 6 circles, their diameter is not fixed, but the spacing between circles need to be identical. As the circles are moved up and down their ...
2
votes
0answers
18 views

Analytic Center of Convex Polytope

I have a convex polytope defined by $Ax \leq b$. I want to know how to find the "analytic center" of my convex polytope, because my goal is to sample from the polytope using Monte-Carlo Markov ...
0
votes
2answers
27 views

Show that an order relation can be defined for the set of points $(x,y)$ of a coordinate plane.

I Think I have to show that the following two axioms hold, I have already shown that multiplication of ordered pairs can be defined (as well as other axioms) showing that it is a field. Although I ...
0
votes
1answer
18 views

new plane equation after transformation of coordinates

I have a plane equation $ax + by + cz + d = 0$ w.r.t to a particular coordinate frame. this coordinate frame w.r.t to the world coordinate frame is $$\begin{vmatrix} r_1 & r_2 & r_3 & ...
0
votes
1answer
35 views

equally spaced on circle question

Define $$\|\vec{x}\|:=\sqrt{\alpha^2+\beta^2},$$ where $\vec{x}:=(\alpha,\beta)\in \mathbb{R}^2.$ Set $$\mathbb{S}^1:=\{\vec{x}\in \mathbb{R}^2: \|\vec{x}\|=1\}\quad \quad and\quad \quad ...
2
votes
0answers
29 views

For what hexagon size can I pack $n$ hexagons into a rectangle of $s$ area?

I have a fixed number of identical regular hexagons I use to build a honeycomb looking grid of hexagons. I have a rectangular container of known dimensions. My job is to figure out how big the ...
1
vote
1answer
14 views

Finding the equation for the tangent plane to earth given latitude and longtiude

I'm creating a program where I need to calculate the equation of the plane tangent to the earth at a given latitude and longitude. I used Projecting an Arbitrary Latitude and Longitude onto a Tangent ...
0
votes
3answers
17 views

Determine if 2 points are horizontal without trigonometry

Let's say that I have 2 points: (c1X, c1Y) and (c2X, c2Y). I would like to consider these 2 points horizontal as long as their angle is below 45 degrees. I could accomplish this with trigonometry. ...
2
votes
0answers
47 views

What's a good book on geometry to read after Kiselev?

I have finished reading both books on geometry by Kiselev and now look to move on but can't find any book to let me do so. Which book would you suggest that one may read after finishing Kiselev?
0
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0answers
33 views

Demonstrative geometry around the world and its significance.

This is not exactly a mathematical question. I am from Pakistan; and over here students are taught a subject 'demonstrative geometry' (as a part of mathematics) from secondary level education. ...
2
votes
4answers
73 views

Find $x$ in the triangle

the triangle without point F is drawn on scale, while I made the point F is explained below So, I have used $\sin, \cos, \tan$ to calculate it Let $\angle ACB = \theta$, $\angle DFC = \angle ...
0
votes
1answer
44 views

A triangle ABC with the internal bisector of $\angle A$, the median drawn from B and the altitude drawn from C meet at the same point.

A triangle $ABC$ with the internal bisector of $\angle A$, the median drawn from $B$ and the altitude drawn from $C$ meet at the same point. Prove that $$\tan A = \dfrac{\sin C}{\cos B}$$ I try to ...
2
votes
0answers
31 views

Translate a geometric theorem into polynomial equations — Theorem of the orthocenter of a triangle

This is Exercise 13 of Chapter 6 of Ideals, Varieties, and Algorithms by Cox et al. The problem asks to translate the following geometric theorem into polynomials and using Groebner basis to test ...
1
vote
1answer
40 views

Geometric problem based on angle bisectors

I am not asking a question,i just want to conform,is my method of solving problem correct? Given a triangle ABC.It is known that AB=4,AC=2,and BC=3.The bisector of angle A intersects the side BC at ...
1
vote
2answers
99 views

Find the value of $h$ from a Kepler-type equation

$$V = \frac{0.5r^{2}\cdot \cos^{-1}(\frac{r-h}{r})\cdot 2-\sin\big(\cos^{-1}(\frac{r-h}{r})\cdot 2\big)}{10^{6}}\tag1$$ This is the equation to find the volume of liquid in a tank in the shape of a ...
1
vote
1answer
51 views

Given an acute triangle ABC with altitudes AH, BK. Let M be the midpoint of AB

Given an acute triangle ABC with altitudes AH, BK. Let M be the midpoint of AB. The line through CM intersect HK at D. Draw AL perpendicular to BD at L. Prove that the circle containing C, K and L is ...
2
votes
1answer
84 views

Is the area of this pentagon $4-\sqrt 5$?

Consider a regular pentagon with vertices (in clockwise order) $A, B, C, D, E$, let $A'$ be the point of intersection of $BD$ and $CE$, let $B'$ be the point of intersection of $CE$ and $DA$, and ...
0
votes
2answers
22 views

Similarity conditions of two right trapezoid with similar angles

We have $2$ right trapezoid for example two trapezoid with angles $90^{\circ},90^{\circ},80^{\circ},100^{\circ}$. do we need to all the sides proportionality or less is enough ?
1
vote
2answers
60 views

Is the Dikin Ellipsoid actually a ball?

I have the inequality (row wise): $Ax \leq b$ The Dikin ellipsoid centered at $x_0$ with radius $r$ is: $$\{z \quad | \quad (z-x_0)^T(z-x_0) \leq \frac{r^2}{H(x_0)}\}$$ where, $$H(x_0) = \sum ...
0
votes
0answers
25 views

Find distance between a plane and some points [closed]

Consider points $x_1,\ldots,x_n$ and plane $w\cdot x-\gamma=0$ in $\mathbb{R}^n$ and let $A=[x_1,x_2,\ldots,x_n]^T$. Is correct following formula to find the distance between these points and the ...
2
votes
2answers
57 views

Question based on triangle inscribed in unit circle

$ \bigtriangleup ABC $is inscribed in a unit circle.If angle bisectors of internal angles at A,B and C meet the circle at D,E and F respectively then value of $\frac{AD \cos\frac{A}{2}+BE ...
0
votes
1answer
24 views

Intersection of Cut Locuses

If $C_p(M)$ is the cut locus of some $p\in M$ in some Riemannian Manifold $M$, then when is: \begin{equation} \bigcap_{p\in M} C_p(M)=\emptyset\text{ ?} \end{equation}
3
votes
1answer
85 views

Prove that $\tan\alpha =\tan^{2}\frac{A}{2}.\tan\frac{B-C}{2}$

Given a triangle ABC with the sides $AB < AC$ and $AM, AD$ respectively median and bisector of angle $A$. Let $\angle MAD = \alpha$. Prove that $$\tan\alpha =\tan^{2}\frac{A}{2}\cdot ...
0
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0answers
19 views

Find the best trapezoidal fitted in an irregular shape [closed]

I am working with some earth irrigation canals. Irrigation canals are usually in trapezoidal shape. These trapezoidal canals are defined by the width of bottom of canal (B) and high of depth of canal ...
1
vote
3answers
73 views

what is the definition of cosine , sine [duplicate]

I know that sine is the ratio of the perpendicular to the hypotenuse of an acute angle. Similarly cosine is the ratio of the base and hypotenuse . But now I found that there is sine and cosine of an ...
4
votes
0answers
59 views

How is it that $\pi$ appears in so many formulas that seem to be in no way geometric. [duplicate]

When I first saw: $$\frac{\pi}{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}\mp\cdots.$$ I was puzzled that such an expression could have anything to do with circles. There are tons more and ...
5
votes
2answers
44 views

Find a plane with distance $3$ from $3x-y-z = 0$

I need to find a plane such that its distance from the plane $3x-y-z = 0$ is $3$. Since distance is defined only for parallel planes, I already know that they have to be parallel, and then, the ...
0
votes
0answers
26 views

Is there a theory for cellular automata propagating signals in straight lines?

Is there a theory explaining how a cellular automata can propagate signals in straight lines? For example, this video shows how some "signals" travel down at a diagonal, even though they are composed ...
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votes
0answers
44 views

Beyond Pythagoras [closed]

Draw an arbitrary triangle $\triangle ABC$. Measure its sides. Draw a ray, $BC$. Draw a circle with radius $AB$. Find the point of intersection, $D$. Measure segment $\overline{CD}$. ...
2
votes
2answers
67 views

Triangle Geometry and Circles Problem

I have discovered something using Geogebra and I am positive it is true. I have tried to prove and my solution works but it is extremley convoluted. I'm hoping someone can provide a simple proof of ...
-2
votes
5answers
44 views

co-ordinate geometry question 3 [closed]

Find intercept of the line whose intercept of $x$-axis and $y$-axis are respectively twice and thrice of those by the line $3x+4y=12$ ?
1
vote
0answers
23 views

Symmetry of stereographic projections of tangent vector to $S^2$ at equator

There is a vector lying in the tangent plane to a sphere $S^2$ at equator. We take two its "stereographic" projections - one from the south pole and other - from north. Projections to the planes ...