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 dim riemann manifold

I am looking for a proof of the thm that any 2d reimann manifold is conformally flat in case of a metric of sig. 0 following thru with prob. 6.30 in text by d'inverno. Thanks.
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1answer
41 views

Evaluate $\iint dydx\;,\bigg\{\frac{\pi}{3}\leq\theta \leq\frac{2\pi}{3}\bigg\}$

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
33 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
27 views

Finding matrix representation of an Ellipsoid

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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21 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 ...
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39 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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12 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
30 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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28 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
29 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
23 views

Farthest point on parallelogram lattice

On points arranged in a parallelogram grid, 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 grid. Or ...
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1answer
11 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
70 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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35 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
28 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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15 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 ...
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2answers
28 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
7 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
22 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
52 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
59 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
44 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
17 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
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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
52 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
23 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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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
787 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
16 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
28 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
13 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?
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32 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. ...
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4answers
71 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 ...
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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
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0answers
30 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 ...
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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 ...
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1answer
50 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
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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 ...
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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 ?
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59 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 ...