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

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1answer
33 views

Does $(x,y,z) = (2,1,1) +s(-1,-1,-1) + t(2,-2,-2)$ represent a line or plane?

Does the equation $$(x,y,z) = (2,1,1) +s(-1,-1,-1) + t(2,-2,-2)$$ represent a line or plane? I claimed it is a plane, as the two direction vectors are not multiples and thus for any values of $s$ and ...
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1answer
19 views

Relation between the lenghts of unequal chords and their distances from the centre

Is there any proportional relation between the lenghts of unequal chords and their respective distances from the centre of a circumference?
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2answers
52 views

Find intersection point of 3 circles

so first of all, I just want to point out that I am a beginner, so cut me some slack. As the title says I have 3 circles. I know the coordinates of each center and the radius of each circle. What I ...
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3answers
49 views

Prove a parallelogram inside parallelogram

I have drawn a figure, In parallelogram ABCD, AP is the bisector of angle A CQ is the bisector of angle C Can I prove APCQ is a parallelogram? or it isn't? I first joined AC and now if somehow I ...
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2answers
36 views

I need some help with Geometry. Is this a correct answer to this problem?

Good day, I have a question regarding geometry. I don't know whether my answer is correct because the answer in my book uses a totally different method for solving this particular problem. Here's ...
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0answers
31 views

Variation of the opaque forest problem (a.k.a farmyard problem)

I was wondering about the following variation of the opaque forest problem (see here and there for previous questions) : What is the least length set of segments that will intersect every straight ...
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3answers
42 views

Distance of centroid to incenter

Suppose there is a right triangle where all side-lengths are integers. The distance from the circumcenter to the centroid of the triangle is 6.5. Find the distance from the centroid to the incenter ...
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1answer
34 views

Simultaneous equation with summation and square - how to solve?

$\mathbf{p}$ is a vector with dimension: $x \times 1$ $\mathbf{d}$ is a vector with dimension: $1 \times y$ $\mathbf{V}$ is a matrix with dimension: $x \times y$ $y \geq x$ $\mathbf{d}$ and ...
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1answer
31 views

Dividing Up A Circular Search Area

BACKSTORY: I need to collect 500 plant samples for strontium analysis. The samples are randomly distributed across a circular area with a radius of 300 kilometers. I have to do this in 30 days, so I ...
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0answers
28 views

Specifications for concave polygons

What makes a polygon concave or convex? We all know that convex polygons have angles less than $180^{\circ}$ and concave polygons don't, but what really makes a polygon concave? In other words, what ...
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1answer
34 views

Length of hypotenuse of a right triangle when dimensions are not scaled equally

What I ask is if $1$ meter in $x$ direction is $2$ times bigger than $1$ meter in $y$ direction. What is the length of hypotenuse when for ex, $3$ in $x$ direction and $4$ in $y$ direction ? I ...
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0answers
47 views

high of a parabola bridge [closed]

My problem is: A bridge has the shape of a parabola with the equation $$x^2=-48y$$ and its arc length is $$l=24m$$ How to compute the heigh of the bridge without using integrals? I mean with an ...
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0answers
16 views

is this sufficient to define a simplex?

I want to define a simplex based on the following properties A convex polytope All vertexes share an edge with all others For a given vertex $v_i$ the set of all facets that the vertex belongs to ...
2
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1answer
105 views

Which of the $43,380$ possible nets for a dodecahedron is the narrowest?

I want to fit multiple regular dodecahedron nets on to an infinitely long roll of paper. I want this to result in the largest possible dodecahedrons, for a roll of a given width. My hunch is that the ...
2
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0answers
33 views

How to calculate the volume of an arbitrary pyramid without calculus?

I've been reading about the intuition behind calculating the volume of a pyramid by dividing the unit cube into 6 equal pyramids with lines from the center of the cube and it makes sense since all ...
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1answer
31 views

Find sides of a right triangle given hypotenuse c and area A (no numbers given)

I've solved couple of these, but I have no idea how to solve it without any numbers provided. I've tried using $A=\frac{ab}{2} \Rightarrow 2A=ab \Rightarrow 4A^2=a^2b^2$ and incorporating ...
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1answer
23 views

Regular Triangulations of Cube

I want to figure out which triangulations of the cube (i.e., partitions into tetrahedra using only the $8$ given vertices) are regular, but I'm not sure how to easily tell whether a given ...
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0answers
49 views

Creating Fundamental Groups and How are they Described?

Im a math learner so this question may seem obvious. Consider the fundamental group of a torus (we call this the object $O(1)$). Suppose we have another torus and can glue it to the other torus to ...
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0answers
19 views

Looking for a conceptual, detailed plane geometry book [closed]

I have a university entrance exam in a few months and i've been having problem with my geometry, the textbooks i own are utterly useless and aren't conceptual in any manner. subjects include: ...
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2answers
45 views

Show that among all triangles with fixed $s$ and $a$, the area is maximised when $b=c$.

Given a triangle $ABC$, let $a= \bigl| BC \bigr|$, $b= \bigl| AC \bigr|$ and $c= \bigl| AB \bigr|$ and let $s=\frac{1}{2}(a+b+c)$ be the semiperimeter. (a) Show that among all triangles with fixed ...
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0answers
32 views

Prove $\frac{3}{64}(ab+bc+ca)^3\geq (de)^3+(ef)^3+(fd)^3$ where $a, b, c$ are three sides of and $d, e, f$ three angle bisectors of a triangle.

A triangle has sides $a, b,c$ and angle bisectors $d, e, f$ where each pair of $a$ and $d$, $b$ and $e$, $c$ and $f$ intersect. Prove that $\frac{3}{64}(ab+bc+ca)^3\geq(de)^3+(ef)^3+(fd)^3$. I was ...
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0answers
12 views

problem on conical frustum [closed]

A glass of conical frustum shape has its two radius 5 cm and 2 cm and height of the frustum is 10 cm.. that means it has a capacity of 408.2 ml. now if we fill the glass up to 200 ml then find the ...
2
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1answer
39 views

${2\pi \over 3} = 2u + \sin {2u}$ (intersections of circles)

So, I was browsing the internet today, when I saw an interesting problem: Two circles, each with radii of one, are intersecting. If the area enclosed by the intersection of the two circles is equal to ...
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1answer
46 views

Question on Geometry, triangles and circles

Let $ABC$ be a right-angled triangle with $B = 90^\circ$ and let $BD$ be the altitude from $B$ on to $AC$. Draw $DE$ perpendicular to $AB$ and $DF$ perpendicular to $BC$. Let $P, Q, R$ and $S$ be ...
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2answers
24 views

Relation between areas of two regular polygons, one with twice as many sides [closed]

I am stuck with the following exercise given on Spivak's Calculus (chapter 8, exercise 11-b): Suppose $P$ is a regular polygon inscribed inside a circle. If $P'$ is the inscribed regular polygon with ...
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1answer
23 views

Pairwise tangent circles radical axes

Three pairwise tangent circles are drawn with the three common tangents to each of the pairs of circles. Prove that the common tangents must intersect at a point. Since the tangents to the ...
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1answer
44 views

Why is it called $SSS$ similarity?

Are two triangles with two sides in proportion automatically similar? If so, why is the postulate called $SSS$ similarity?
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2answers
25 views

Find the sum of the areas of all rectangles whose area is tripled when three units are added to the height and two units are added to the length

A rectangle has all sides of integer length. When three units are added to the height and two units to the length, the area of the rectangle is tripled. What is the sum of all the original areas of ...
1
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1answer
31 views

Calculating cross section of rectangle by angle

Given this: How do you calculate the length of the green line, given x degrees, and the fact that height / width = 2 / 5? The blue line indicates at 0 degrees. The length of the pink line equals ...
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1answer
29 views

Prove that $\angle I_aB_0I_c=90$

Given triangle $ABC$ and point $D$ on $AC$. Let $I_a, I_c$ be the centers of inscribed circles of $ABD$ and $BCD$ respectively. $B_0$ is the point, incircle of $ABC$ touches $AC$. Prove that ...
3
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1answer
44 views

Is the matrix filled with the areas of pairwise intersections of disks in a plane always positive semidefinite?

Consider disks $s_1, \cdots, s_n$ in the plane and let $a_{ij}$ be the area of $s_i\cap s_j$. Is it true that for any real numbers $x_1,\cdots, x_n$ we have $$ \sum_{i,j=1}^n x_ix_j a_{ij} \geq 0$$ ...
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0answers
21 views

What is the solid of revolution of an ellipsis around a general line through the origin?

If one rotates an ellipsis around its major axis, one gets a prolate ellipsoid. Around the minor axis, ones gets an oblate ellipsoid. What about a general line through the origin?
2
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1answer
58 views

Motivation for studying rational curves

Why do we study rational curves? A curve $f(x,y)=0$ is called a rational curve if there exists two rational functions $\chi(t)$ and $\psi(t)$ such that $f(\chi(t),\psi(t))=0$ for all $t$. Why is it ...
3
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2answers
31 views

What types of triangles are constructible?

What types of triangles are constructible? I know that equilateral triangles are easily constructed using compass and straightedge, but what about other types of triangles? Can any other ...
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4answers
24 views

Feynman lectures, Volume I, chapter 13-4

While reading Feynman lectures on Physics, volume I, Chapter 13-4, I found following assumption, which I don't understand: Then, since $r^2 = \rho^2 + a^2$, $\rho\,d\rho = r\,dr$. Therefore ... ...
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0answers
30 views

What type of math uses congruence in spheres and was what I proved true?

I used a good portion of my summer break to read about topology and I'm interested to know if what I proved was informally a method used in topology? If not what type of math did I employ to prove ...
0
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1answer
10 views

How to prove that the pointreflection at the midpoint of two several points out of a regular pointlattice fix the lattice?

How to prove that the pointreflection at the midpoint of two several points $A,B\in\mathfrak{L}$ in a regular pointlattice $\mathfrak{L}$ fix the lattice $\mathfrak{L}$? We call ...
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2answers
48 views

Finding the equation of a circle from the equation of its tangents

Given the equation of a pair of lines : $36x² - 63xy + 20y² + 54x - 17y - 10 =0.$ If the circle touches one of the lines at (-3,-1) and the other at some point then find the equation of the ...
2
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1answer
53 views

Integer length right triangle with hypotenuse $2^{100}$

Is it possible to construct a right triangle with integer side lengths and a hypotenuse of $2^{100}$? After looking at a list of pythagorean triples, I couldn't find a hypotenuse of a right ...
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1answer
28 views

Calculate areas from a sliced trapeze

I have an area shown on this picture for $n=5$. Total area always equals 1. I can slice the area to $n$ pieces, every piece have the same height which equals $\frac2n$, $x_0$ and $x_n$ are known ...
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2answers
92 views

Erdős-Mordell theorem geometry proof

Using the notation of the Erdős-Mordell theorem, prove that $PA \cdot PB \cdot PC \geq \dfrac{R}{2r}(p_a+p_b)(p_b+p_c)(p_c+p_a)$. The notation of the Erdős Mordell theorem means that $p_a$ for ...
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0answers
20 views

what is special about simplexes?

Is it true that the $d$-simplex is the only convex polytope such that All vertexes share an edge with all others, It is transitive under the action of a group?
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1answer
44 views

Hexagon and incircle of triangle inequality.

Consider the three lines tangent to the incircle of a triangle $ABC$ which are parallel to the sides of the triangle; these, together with the sides of the of the triangle, for a hexagon $T$. ...
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2answers
23 views

Finding the error in the surface area of a cube. when length = 3, error= ${1\over 4} $

Find the approximate error in the surface area of a cube having an edge of length 3ft if an error of ${1 \over 4}$ in. is made in measuring an edge I have to do this by using differentials and ...
0
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1answer
18 views

What does the objective function compute when trying to find the maximum distance between two supporting hyperplanes?

I am reading this paper about Support Vector Machines and need clarification on the method used to maximize the distance between two supporting hyperplanes. First, some definitions: Let: $A$: an ...
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1answer
14 views

In Quadrilateral inscribed $ABCD : (AB*CD) + (AD*BC) = AC*BD$

IF we have a Quadrilateral inscribed $ABCD$, then $(AB*CD) + (AD*BC) = AC*BD$.
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1answer
47 views

A problem with “Crossed Ladders Theorem”

In the following diagram, $AY ||BZ$, $AB$ is base. $M$ is $5$ above $N$ and $N$ is $4$ above $O$. What is the height of the triangle $\Delta AOB$. My Work There is a theorem named Crossed Ladder ...
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0answers
12 views

Sphere inside cylinder vs polyhedra?

Comparing a cylinder with a polyhedra that has a symmetric coxeter $\ge 3$. Both have their centers hollowed out by $k\%$, in the shape of their outer, i.e.: relative to top face Which can better ...
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3answers
44 views

Inside an not Equilateral Triangle what is the sum of distances from a random point to 3 sides

Given an not Equilateral Triangle with following side sizes: 45,60,75. Find a sum of distances from a random located point inside a triangle to its three sides. Note 1: Viviani's theorem related only ...
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1answer
44 views

2011 AMC 12A #13 — Different answers to triangle geometry problem

Triangle ABC has side lengths $AB = 12$, $BC = 24$, and $AC = 18$. The line through the incenter of triangle ABC parallel to $\overline{BC}$ intersects $\overline{AB}$ at M and $\overline{AC}$ at ...