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

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Segments of a hypotenuse

The hypotenuse of a right triangle is divided into 2 segments by the altitude to the hypotenuse The sum of the greater segments on the hypotenuse of 2 disimilar right triangles is equal to the ...
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2answers
29 views

Can you do this to find circumference from area of a circle

If you divide the circumference by $2$, does it equal the area divided by the radius? That is, do you have $C/2 = A/r$ for any circle? ...
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2answers
8 views

Geometrical calculation to enlarge the height of rotated rectangle

There is a polygon (rotated rectangle) that defined by 4 corner points in 2D coordinate system. Does anyone help me with the fast (minimum trigonometry operations) algorithm to change its height by ...
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0answers
25 views

Minimum perimeter of a triangle when two altitudes are known [on hold]

Two altitudes of triangle $\triangle ABC$ perpendicular to the $a$ and $b$ sides are given as $u,v \gt 0$ respectively. Can we express the minimum perimeter of such $\triangle ABC$ only in terms of ...
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0answers
29 views

Maximum height of the side $b$ of any $\triangle$ known perimeter and height of $a$ side [duplicate]

Get the formula only with the data. The maximum height corresponding to the side $b$ of any $\triangle ABC$ once known the value of its perimeter and height corresponding to the $a$ side.
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1answer
19 views

Polarity on a Hyperboloid of one sheet

Given a quadric $Q = \{v \in \mathbb{R}^n \mid \alpha(v,v) = 1\} \subset \mathbb{R}^n$, defined by a bilinear form $\alpha: \mathbb{R}^n\times\mathbb{R}^n \to \mathbb{R}^n$, and an affine subspace ...
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1answer
31 views

'Chasing sides' in a geometry problem

Consider the circle $W=x^2+y^2=81$. Let $AB$ be a diameter of circle $W$. $AB$ is extended through $A$ to $C$. Point $T$ lies on $W$ so that line $CT$ is tangent to $W$. Point $P$ is the foot of the ...
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0answers
20 views

Intersection of Random Subspace and Hypercube

Suppose that $A \subset \mathbb{R}^n$ is a random linear subspace of dimension $k < n$. I am interested in the event that $A$ intersects the hypercube $[-1,\ 1]^n$ at a specific face. In other ...
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0answers
31 views

Triangle inscribed in another triangle

If $a,b,c$ are the sides of a triangle,$\lambda a,\lambda b,\lambda c$ the sides of a similar triangle inscribed in the former and $\theta$ the angle between the sides $a$ and $\lambda c$,prove that ...
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0answers
8 views

2 dimension riemann manifolds of signature 0 metric

Does anyone have a proof that any 2d riemann manifold is conformally flat if metric has signature 0? Thanks.
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1answer
57 views

Evaluate $\iint dydx$ on the domain $0\leq r\leq1$, ${\pi}/{3}\leq\theta \leq{2\pi}/{3}$ [duplicate]

I need to evaluate $\displaystyle\iint \color{red}{dydx}\;\;\;,\bigg\{\frac{\pi}{3}\leq\theta \leq\frac{2\pi}{3}\bigg\}\;\;\;\;,0\leq r\leq1$ $\color{blue}{\text{without using polar ...
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4answers
43 views

Finding the equation of the straight line $y=ax+b$?

If I have a circle $x^2+y^2=1$ and line that passes trough $(0,0)$ and I know the angle between the line and the axis. If, for example, the angle is $\frac{\pi}{3}$, how can I find the equation of ...
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2answers
33 views

Finding matrix representation of an Ellipsoid [on hold]

I have a $2$-dimensional ellipsoid centered at $(1,2)$. The axes are parallel to $y=x$ and $y=-x$, and it passes through points $(-1,0)$, $(3,4)$,$(0,3)$,$(2,1)$. I would like to find the symmetric ...
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0answers
25 views

Rotating one coordinate system about another

I have two coordinate systems: A and B. I also have a point p, whose position relative to ...
3
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0answers
51 views

What is the name of this shape? (spacetime)

After seemingly endless searching for terms such as curved cone, hyper-cone etc I am at a loss as to what this shape is called. I believe it is commonly used to depict the curvature of space time. ...
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1answer
21 views

Clarify formula that computes number of dies on wafer

I want to compute the number of dies per wafer (also DPW in the following). There are some formulas, that can be used to do so: ...
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1answer
36 views

A good book on basic (Euclidean) geometry.

We were studying demonstrative geometry, so I thought if I read Euclid's Elements it would give me the proper conceptual basis to understand the theorems. But then I learned that Euclid's method of ...
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0answers
29 views

Number of triangles possible in android lock patterns?

I recently starting using the patternlock on my android phone and i play around with it a lot, just drawing lines until im locked out for 30 secs. I thought i'd make it into a pointless game of ...
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3answers
38 views

Angle between medians in right triangle

In a right angled triangle,medians are drawn from the acute-angles to the opposite sides.If maximum acute angle between these medians can be expressed as $tan^{-1}(\frac{p}{q})$ where p and q are ...
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2answers
43 views

Farthest point on parallelogram lattice

On points arranged in a parallelogram lattice, like on the image in this Wikipedia article, how to calculate the maximal distance any point on the plane may have to its closest point from the lattice. ...
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1answer
18 views

How to show the average x-coordinates of four collinear points on the curve is a constant?

Show that if four distinct points of the curve $y=2x^4+7x^3+3x-5$ are collinear then their average x-coordinate is some constant k. Find k. Shall I use vector to calculate their x-coordinate, or ...
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4answers
72 views

How to prove that the line perpendicular to the radius is the tangent in the calculus sense?

Let $P=(p_1,p_2)$ be a point on an semicircle and $r$ be the line perpendicular to the radius $\overline{OP}$, like the picture below. Euclid showed (Book III, Proposition 16) that $r$ does not ...
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0answers
41 views

“Maximum point lies on a curve” implies tangential derivative is zero there.

Given a differentiable function $f:\mathbb{R}^2\to\mathbb{R}$, suppose that it has a local maximum at the point $(x_0,y_0)$. Let $\gamma$ be a smooth curve passing through $(x_0,y_0)$. Does it follow ...
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2answers
30 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. ...
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0answers
18 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
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2answers
29 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 ...
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1answer
9 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. ...
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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 ...
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2answers
53 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
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1answer
66 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 ...
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1answer
46 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 ...
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1answer
22 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.
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2answers
35 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 ...
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1answer
18 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 ...
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2answers
44 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 ...
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2answers
32 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 ...
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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
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2answers
54 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 ...
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2answers
24 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 ...
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0answers
28 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 ...
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4answers
790 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 ...
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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
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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 ...
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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 ...
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1answer
17 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 & ...
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1answer
32 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
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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 ...
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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 ...
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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. ...
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0answers
40 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?