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

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0
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0answers
11 views

Find the length of 2 points based on intersection of a circle

Would anyone know the formula for finding the length of BC giving the below data. AB = 20 r = unknown BC = ? The other dimensions that can be used if needed are written on the diagram. Thank ...
3
votes
1answer
32 views

Necessarily a homeomorphism?

Let $D$ be the projective curve defined by $y^2z = x^3.$ Consider the map $f: \mathbb{P}_1 \to D$ defined by$$f[s, t] = [s^2t, s^3, t^3].$$Is it necessarily a homeomorphism? Any help would be greatly ...
1
vote
1answer
21 views

how many possible acute triangles with perimeter given

How many possible acute triangles exist with perimeter 18? All sides are positive integers. The triangle (7,7,4) is the same as (4,7,7). I need the work in a way that a geometry 9th grade student ...
2
votes
0answers
49 views

Geometric proof for Sophomore's dream

Is there a "visual proof" for sophomore's dream? $$\int_0^1 x^{-x}\,dx = \sum_{n=1}^\infty n^{-n}.$$ In the wikipedia article there are two algebraic proofs, but the integral and the summation has ...
1
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0answers
13 views

Find parametric line between two 2D line segments that is an exact distance from a point

Given two 2D line segments, $\overline{ab}$ and $\overline{cd}$, and a point $p$, I would like to find a scalar value $t$ such that the line segment between $\overline{ab}(t)$ and $\overline{cd}(t)$ ...
0
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0answers
21 views

The degree-genus formula cannot be applied to singular curves in $\mathbb{P}_2$?

(The degree-genus formula) The Euler number $\chi$ and genus $g$ of a nonsingular projective curve of degree $d$ in $\mathbb{P}_2$ are given by$$\chi = d(3-d)$$and$$g = {1\over2}(d-1)(d-2).$$ My ...
1
vote
1answer
17 views

Can a compact set of $\mathbb{R}$ have some properties and not being convex

The question is related to this one On a condition when bounded sets in R n is convex ?. Suppose that $n > 1 $ and that $C \subset \mathbb{R}^n$ is a compact (closed and bounded) set having a ...
1
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1answer
26 views

What will be the equation of side $BC$.

The equation of two equal sides $AB$ and $AC$ of an isosceles triangle $ABC$ are $x+y=5$ and $7x-y=3$ respectively . What will be the equation of the side $BC$ if the area of the triangle ...
6
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6answers
207 views

Finding the largest triangle inscribed in the unit circle

Among all triangles inscribed in the unit circle, how to find the one with the largest area?
2
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2answers
34 views

The four straight lines given by the equation $12x^2+7xy-12y^2 =0$ and $12x^2+7xy-12y^2-x+7y-1=0$ lie along the side of the?

I know these equations are called general equation of second degree and also represent a pair of straight lines . I could extract lines from the equation $$12x^2+7xy-12y^2 =0 $$( these are $$ ...
3
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1answer
43 views

length of the curve $y=x^n$ in the unit square

Let $l_n$ be the length of the curve $y=x^n$ in $[0,1]\times[0,1]$. Then obviously $\lim_{n\to\infty}l_n = 2$. What about $\lim_{n\to\infty}(n(2-l_n))$ ? The formula $l_n = ...
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1answer
42 views

How to do this triangle question? [on hold]

ADB is a straight line. Prove that AD = BC
3
votes
1answer
281 views

Prove this is an isosceles triangle

In a triangle ABC, $\sin B\cdot\sin C=\cos^2(\frac{A}{2})$ Prove that this is an isosceles triangle. Can anyone guide me to prove this? Thanks
2
votes
2answers
30 views

sum of perpendicular distances from the sides of a triangle.

I am trying to solve a problem and got stuck in the following:- P, A’, C’ are respectively points on the sides AC, CB, and AB of ⊿ABC. PA’ and PC’ are the perpendiculars to the sides of the ...
0
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1answer
29 views

Check if the following are perpendicular.

I have these expressions : $$2x+2y-5=0 \\ x=3-t,y=2+t,z=1-3t$$ I need to check if they are perpendicular. This is what I did : The following vectors represent the expressions $\langle ...
1
vote
2answers
13 views

How to find the vertex of a rhombus?

I am unable to solve this question. If the area of a rhombus is 10 sq.unit . It's diagonals intersect at (0,0) if one vertex of the rhombus is (3,4) , then one of the other vertices can be ? I took a ...
2
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1answer
68 views

Trirectangular tetrahedron

Looking at http://mathworld.wolfram.com/TrirectangularTetrahedron.html I wonder what the symmetry group of a trirectangular tetrahedron is?
4
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4answers
37 views

Are circles and lines in two-space one-dimensional?

Circles and lines are normally regarded as one-dimensional objects. However, when embedded in two-space, they require two coordinates $(x,y)$ to specify a point within them. Are they still considered ...
1
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0answers
37 views

Geometry of Curves

I found this question in question paper of Geometry of Curves and surfaces from Leeds University. Can anyone help me how I solve it.
0
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1answer
13 views

Sacle the distance of lattice points

I know that for a hexagonal lattice generated by (0,1) and ($\sqrt{3}/2$,1/2) (i.e., when the distance between lattice points is 1), the number of lattice points in a circle of given radius $r$ can be ...
3
votes
2answers
33 views

If $x-2y+4=0$ and $2x+y-5=0$ are the sides of isosceles triangle having area $10$ sq unit .Equation of third side is?

Okay, I know two sides of an isosceles triangle are equal . I have also taken out the intersection points of the lines given in the question . Other than this , I have no clue about how I will find ...
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3answers
19 views

Complex plane (Show that triangle is right-angled)

The points $O$,$P$ and $Q$ in the complex plane represent the complex numbers $0+0i$, $4+2i$ and $3-i$ respectively. Find the exact length of $PQ$ and hence, or otherwise, show that triangle $OPQ$ is ...
3
votes
6answers
97 views

Visualize $z+\frac{1}{z} \ge 2$

As we know, always $$z+\frac{1}{z} \ge 2,~~~~~~~~~ z\in \mathbb{R}^+$$ However, is there any geometric way to visualize this equation for some one who is not that expert in math? I know this ...
6
votes
3answers
141 views

Difficult Coordinate Geometry and Calculus Question

I was given this question by a friend and after tirelessly working on it I have not come up with anything substantial. I was hoping someone in the community could provide a pointer or possibly a ...
-1
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0answers
21 views

how t calculate the area of a rectangle intersected by an arc [on hold]

I need to make two calculations: Determine the arc that is needed between the two intersection points of a rectangle. Knows: Size of the rectangle, location of the intersection points. I want to ...
2
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0answers
44 views

What's the most general geometry branch?

What is the most general geometry of curves and surfaces? For example, at curves, we define in differential geometry the tangent vector as the derivative of a regular curve, but visually many other ...
2
votes
1answer
29 views

Calculate tangent points of two circles.

I have 2 circles with given center coordinates and radius. And now I need to find the coordinates of all 8 tangent points to those circles? I found this site explaining exactly what I want do to: ...
0
votes
2answers
43 views

What type of angle is $3+ \frac{1}{6}$ of a complete rotation?

Angle less than $90$ deg is acute, angle greater than $90$ and less than $180$ is obtuse and angle greater than $180$ deg is reflex. Now, what if an angle is a $3+\frac{1}{6}$ of a complete rotation? ...
0
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0answers
11 views

Angle for sloped surface intersection

I'm not sure if this is the right SE site for this questions so apologies if it's not. But I've got a question about calculating angles that I can't seem to figure out the maths for. To add a bit of ...
0
votes
2answers
18 views

Which of the following point is outside the triangle?

If $P(6,7),Q(2,3)\ \text{and}\ R(4,-2)$ be the vertices of the triangle , then which of the point is not contained in the triangle? $a.)(4,3)\quad \quad \quad \quad b.)(3,3)\\ c.)(4,2)\quad ...
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0answers
28 views

Where can I find a good drawing software?

Maybe this is a little off-topic but often, when writing articles, I find myself in need of a good drawing software (for MAC or Windows) that would allow me to draw figures like the one below: Do ...
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0answers
36 views

Square & square root [on hold]

What is the minimum distance a snake which can only crawl has to cover t to reach the diagonally opposite vertex of the cube. IF Side and height is "S".
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votes
1answer
41 views

Diagonal of a cube [on hold]

The hyperdiagonal of a cube extends from the upper back left to the lower front right. If all the side lengths of the cube are $6$ inches, what is the length of the hyperdiagonal of the cube?
4
votes
2answers
58 views

The lines $x+2y+3=0$ , $x+2y-7=0$ and $2x-y+4=0$ are sides of a square. Equation of the remaining side is?

I found out the area between parallel lines as $ \frac{10}{\sqrt{5}} $ and then I used $ \frac{|\lambda - 4|}{\sqrt{5}} = \frac{10}{\sqrt{5}} $ to get the values as $-6$ and $14$ . I am getting the ...
2
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0answers
25 views

Prove that AX is symmedian

Let $ABC$ be a triangle and let $M$ be the midpoint of $BC$. Let $O_1$ be the circumcenter of $ABM$ and $O_2$ be the circumcenter of $ACM$. $X$ is the circumcenter of $ABC$. Prove that $AX$ is the ...
1
vote
2answers
37 views

Circumcenter of Tetrahedron (in 4D)

I am trying to calculate the circumcenter of a tetrahedron in 4 dimensional space. Basically what I am looking for is the center of the smallest sphere which passes through all 4 vertices of the ...
0
votes
1answer
14 views

Points nearby a line

Suppose we have a number of points in $2d$. I'm looking for a way to determinate a line, which has a maximum number of points in a given range. There is no need that the line intercepts one of the ...
1
vote
2answers
47 views

Problem on Straight lines

I am working on this question. A light ray coming along the line $3x+4y=5$ , gets reflected from the line $ax+by=1$ and goes along the line $5x-12y =10$. Now, I have to find out the value of $a$ and ...
3
votes
1answer
21 views

The necessary and sufficient condition for a regular n-gon to be constructible by ruler and compass.

I have a problem concerning the necessary and sufficient condition for a regular n-gon to be constructible by ruler and compass. $\bf My$ $\bf question:$ For a given positive integer $n$, how can we ...
2
votes
2answers
36 views

An equation involving ratios in a triangle.

In triangle $ABC$, if the incenter is $I$ and $AI$ meets $BC$ at $D$, show that $$\frac{AD}{ID}=\frac{AB+BC+CA}{BC}$$ I tried using similar triangles and got nowhere, couldn't find any use for the ...
22
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5answers
2k views

I think I see mysterious lines inside triangles—how to prove their existence?

Lately I've been fooling around with points inside a triangle and the sum of their distances from all sides. This was when I noticed a weird behaviour: For each point I chose there always seemed to ...
2
votes
0answers
19 views

Compute volume of the tetrahedron from circumsphere test

I'm working on a computational geometry algorithm. In every iteration I solve the matrix below, where (a,b,c,d) are the vertices of a tetrahedron, and e is an arbitrary point. Solving the determinant ...
0
votes
2answers
27 views

Constructions of perpendicular in hyperbolic plane

Consider the disc model of hyperbolic plane $\mathbb{D}^2$ and a line $g$ through the origin $(0,0)\in \mathbb{D}\subset\mathbb{C}$, i.e. a diameter of the circle $\partial \mathbb{D}=S^1$. Let ...
4
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0answers
48 views

Are closed simple curves with that property necessarily circles?

This is a more interesting follow-up to the question Are closed simple curves with this property necessarily circles? Let $\gamma:[0,1]\to \mathbb R^2 $ be a closed simple $C^1$ convex curve and ...
2
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1answer
49 views

Are closed simple curves with this property necessarily circles?

Let $\gamma:[0,1]\to \mathbb R^2 $ be a closed simple curve and $\Gamma$ be the region enclosed by $\gamma$. Let $O$ be the center of mass of $\Gamma$. Suppose that any line that goes through ...
1
vote
1answer
69 views

What is the name of this geometric shape?

#1 I am trying to find the name for this when $d1 = d2$ What is the name of this object? #2 Assume d1 is different than d2. What is the name of this kind of object?
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0answers
49 views

the formula for the volume

If you know that the volume of cube ($a^3$) represents the sum of the surface squares ($a^2$). Following the same logic: If we know the area of the circular segment is equal to the area of the ...
0
votes
2answers
27 views

The angle between $u$ and $v$ is $30º$, and the vector $w$ of norm $4$ is ortogonal to both $u,v$. Calculate $[u,v,w]$.

The angle between the unit vectors $u$ and $v$ is $30º$, and the vector $w$ of norm $4$ is ortogonal to both $u,v$. The basis $(u,v,w)$ is positive, calculate $[u,v,w]$. I did the following: ...
1
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0answers
32 views

Intersecting lines in sectors of a circle.

Good day everyone, I'm trying to simulate a Laser Range Finder (LRF for short) in a corridor environment. I'm including a small fast sketch I did of this. I can't upload images yet, so I include just ...
1
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2answers
44 views

Dual plot for complex roots of quadratic equation

Real roots of quadratic equation $ x^2 - \sqrt 3 x + 1/2 =0 \tag{1} $ can be plotted on $x$- axis as its parabola intersection at $ (\sqrt 3/2 \pm 1/2,0). $ In an improvization I assign ...