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

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25 views

Geometric interpretation of $x_1^2y_1^2+x_2^2y_2^2+x_3^2y_3^2+\dots$

Say $x$ and $y$ are two $L_2$ unit vectors of size $n$. In that case the inner product: $$x_1y_1+x_2y_2+x_3y_3+\dots+x_ny_n$$ Is the cosine of the angle between them. For an application I was ...
1
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1answer
22 views

Shortest distance between a point and a plane using orthogonal projection

Given a point $v = (x_1,y_1,z_1) \in \mathbb R^3$, and a plane P:= $ax+by+cz=d$, find the shortest distance between $v$ and $P$. My attempt at a solution Consider $U = \{(x,y,z) \in \mathbb ...
3
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0answers
13 views

How prove that $II^{\prime}< AA^{\prime}$ for $I $ and $I^{\prime}$ be their incenters?

Assume that we have two triangles $ABC$ and $A^{\prime}BC$. Let $I $ and $I^{\prime}$ be their incenters. How prove that $II^{\prime}< AA^{\prime}$? I have no idea how to do this, can this be ...
3
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1answer
21 views

The composition of rotations

How to prove that the composition of 2 rotations about an axis $l_1$ and $l_2$ is the rotation? I know that we can represent rotation about the axis $l$ at angle $\phi$ as the composition of 2 ...
1
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1answer
30 views

Align small circles on big circle at same distance

Consider we have a circle with radius $R$ positioned on $O (0, 0)$. Having $n$ small circles, what's the easiest method to get the center points (of the small circles) knowing the $R$ and $n$ values? ...
3
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1answer
77 views

Inequality of length of side of triangle

For any triangle with sides a,b,c $$a^2b(a-b)+b^2c(b-c)+c^2a(c-a)\ge 0$$ I tried substituting $a=x+y$, $b=y+z$, $c=z+x$ but well it doesn't help in any sense except wasting 3 pages that lead to ...
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0answers
34 views

Points and Triangles [on hold]

(Miklós Schweitzer 2014, Kömal November 2014 A. 626) We have $4n+5$ points on the plane, no three of them are collinear. The points are colored with two colors. Prove that from the points we can form ...
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2answers
31 views

solving for sin of the sum of two angles of a triangle

In triangle $ABC$, $3\sin B+4\cos C=6$ and $4\sin C+3\cos B=1$. Show that $\sin(B+C)=0.5$. Can we assume $\angle A = 180 - ( B + C)$ and use sum formula.
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1answer
21 views

Is a closed surface in 3-sphere a boundary of a handlebody? [on hold]

If a closed surface is embedded in 3-sphere, then does it bounds a handlebody?
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2answers
22 views

How to find the inradius of a triangle with given side lengths?

I need to find the inradius of a triangle with side lengths of $20$, $26$, and $24$. I know the semiperimeter is $35$, but how do I find the area without knowing the height? Thank you.
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1answer
43 views

How to calculate the volume of an “icecream cone”?

How to calculate the volume of this shape? I've tried to do $2.5\cdot 2.5\cdot 3.14\cdot 12$ and then divide that with $3$ since the first part is a cone. But I only get $78.5$ by doing that. Am I ...
4
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6answers
336 views

Can two shapes occupy the exact same area on a plane?

Suppose there are two triangles on a plane. The coordinates for each point of both triangles are the same. It seems to me that there is nothing to differentiate these two triangles, and as such, ...
1
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3answers
128 views

How can I find the sine or the tan or the cos of an angle in radian?

There is an angle equal 0.54 radians and opposite leg equal to 3 units, I need to find the length of the adjacent leg. I know that I have to do ${\rm leg} = \frac{3}{\tan(0.54 \text{ rad})}$. I got ...
6
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0answers
51 views

Computing Hodge numbers of a complete intersection

The situation is this: I have a 5-dimensional irreducible projective variety $Y$ embedded in $\mathbb P^{13}$. This variety is singular, the singularities being a disjoint union of two curves. I have ...
1
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1answer
23 views

Rotation of curve function

I am working on some coding where I require expertise in field of Mathematics. I have a function: $$F(x) = -0.007x^4 + 1.971x^3 - 190.4x^2 + 8150x - 13024$$ I want to rotate some section of this ...
1
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1answer
50 views

philosophy : first axiom of geometry and variable curvature

The very first axiom of geometry can be described as: Two different points lay on one and only one line. And I was wondering are there surfaces where this axiom irrecoverably fails? and I found ...
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1answer
14 views

finding the intersections of two polar angles on a segment

You are given a point Q and a segment defined by end points A and B. Given two polar ...
1
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2answers
38 views

7 circles surrounded by 12 circles

In preparation for a math contest my math teacher gave me some interesting math exercises. I am really stuck on the following task: Twelve coins are placed flat on a table so that their centers ...
0
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0answers
28 views

Finding possible 2d positions of a transmitter, using multilateration.

I'm trying to find the possible 2D positions of a transmitter, using 2-3 receivers (I'd prefer 2, I don't need the actual position.) using the math from this Wikipedia site. So far I've only been able ...
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2answers
24 views

Find the area of the triangle using different methods? [on hold]

Find the area of the triangle with vertices $(1,0) ,(2,2) ,(3,1)$ using different methods ?
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0answers
43 views

Complex numbers and geometry - Four complex numbers lying on a circle

I'm stuck on a problem of What is Mathematics?, by Courant and Robbins. The formulation is as follows: Prove that if for four complex numbers $z_1$, $z_2$, $z_3$ and $z_4$, the angles of ...
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1answer
24 views

What is the ratio of areas of quadrilaterals ABFE and EFCD??

$ABCD$ is a trapezium with parallel sides $AB = a$ and $DC = b$. If E and F are mid-points of nonparallel sides AD and BC respectively, then what is the ratio of areas of quadrilaterals $ABFE$ and ...
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0answers
17 views

4m rectangle ABCD a square is cut off to leave rec BCEF. Rec BCEF is similar to ABCD. Find x and hence state the ratio of the sides of rec ABCD.

aoa. m the student of matric and doing IGCSE mathematics by DAVID RAYNER. I M DOING GEOMETRY (similarity). i m facing sme problems in doing this, so i need help for solving these problems
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2answers
35 views

Geometric Series Word Problem from Khan Academy

A Sierpinski triangle, starts with a white equilateral triangle with sides of length $1$. First, the middle triangle is colored green. At the second step, 3 triangles are colored blue. At the third ...
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0answers
35 views

In the regular hexagon tell each area,But How find this length with $B_{i}B_{i+1}$?

in the Figure:Regular hexagon $A_{1}A_{2}A_{3}A_{4}A_{5}A_{6}$,and $B_{1}B_{2},B_{3}B_{4},B_{5}B_{6}$ intercept at $O$,and such any two line included angle of $60$ degrees( mean $\angle ...
2
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3answers
98 views

Can $x^n = 1$ describe the equation of a unit circle?

The complex roots of the equation $x^n = 1$ lie on the unit circle. Suppose $n$ goes large. Is it correct to say that all the roots of $x^n=1$ form a circle of radius one?
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2answers
21 views

A Coordinate geometry problem: [on hold]

How to find a point on $x$-axis,which is equidistant from points $P(0,-4)$ and $Q(3,4)$.
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0answers
42 views

area of triangle xyz by knowing perimeter and square of sides [on hold]

Any help in this problem would be so greatly appreciated: In triangle $XYZ$, if $x+y+z=24$, $x^2+y^2+z^2=200$, find area of triangle $XYZ$ This question should be solved in rational numbers, not real ...
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3answers
105 views

The area of intersection of an isosceles triangle with another triangle

I tried graphing the equations that form the two isosceles triangles and integrating the bounded area and got 7.456 as my answer after rounding. The answer key has the answer listed as 7.2 However, ...
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0answers
66 views

How can I understand trigonometric expressions? [on hold]

Condition of task, I have to find sin, cos, tangent with angles 120 degrees. I have solve $$\sin 120 = \sin (180 - 60) = \sin 60 = \frac{\sqrt{3}}{2}$$ Why $$ \sin 120 = \sin 60 = ...
1
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2answers
88 views

showing the equation is true [on hold]

Am from writing my class test and I have failed to answer this question. None of my friends could help me after the test was over.the question reads: If $P$ is the length of the perpendicular from ...
0
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0answers
33 views

How prove that $\angle$ $CIG=180$ if and only if $GM || AB$?

$M$ is the intersection of medians in $\triangle$ $ABC$ and $I$ is the center of incircle,$A_1$ and $B_1$ are touch points at $BC$ and $AC$,respectively.$AA_1$ meets $BB_1$ at $G$. How prove that ...
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1answer
24 views

How do you use Simpson's rule?

Using Simpson's rule and and interval of d =$0.5,$ approximate the area of the region bounded by the curve defined by $y = $$\sqrt{x+2}$ and the $x-axis$ This is in my geometry book, however there ...
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0answers
30 views

How find the area of the figure $CBOED$? [on hold]

Gien circle and an isosceles trapezium $CBED$ inscribed in it, such as $<EDC=75^o$ $BC || DE$ $DE=6\sqrt{3}$. If $O$ a point inside the circle, such as $OB=BC$. How find the area of the figure ...
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0answers
36 views

difficult question (centroids) [on hold]

i've been strugling with two questions involving centroids: $A,B,C,D$ are four points. 1)- Find the set of points $M$ such that: $\overrightarrow{DM} = ...
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1answer
17 views

Distance from a point to Voronoi hyperplanes

I'm in the process of implementing this paper and I'm running into an issue with the process listed in portion V-A (page 5). Specifically, the paper mentions that I should store the distance from each ...
2
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2answers
57 views

Finding Locus of a Midpoint of a Chord

Let $A$ and $B$ be points, and let $C$ be a point that varies on the open semicircle with diameter $AB$. Construct squares externally on sides $AC$ and $BC$, and let $D$ and $E$ be the centers of ...
0
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1answer
27 views

Understanding normal and binormal of a vector or of a spline

I found a paper where it computes the 3D trajectory of a quadrotor and defines an error position as the difference between 2 vectors (here the source, under 3D trajectory control): $$ e_{p} = ...
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2answers
84 views

Transform polygons into one another?

I am aware that there must be no standard way to achieve this, but I don't know what has been done so far. I feel like I'm missing keywords to investigate further. I have any two 2D polygons $a$ and ...
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0answers
37 views

filling polygons automatically [on hold]

how we can fill polygons automatically like in this example, we have two type of polygons(pink and black) , the first type the black polygons, should grow to occupy the void area between the black ...
0
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1answer
27 views

How to find the farthest point on a known graph from a known line?

How to find the farthest point on a known graph from a known line? For example, farthest point on the graph of $f(x)=x^3$, and the line segment that passes between $(0,0)$ and $(1,1)$
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0answers
23 views

Estimating distance between two points in n-dimensional space, with knowledge of other paths

Suppose there exist four randomly distributed points in $n$-dimensional space: $A$, $B$, $C$, and $D$. We have no knowledge of the coordinates of any of these points, but we do know nearly all of the ...
2
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1answer
34 views

Circle equation solution

Hi I'm stucked with this equation while transforming it into circle equation: equation is $y+\sqrt{x-x^2} = 0$ Here is my solution: $$y+\sqrt{x-x^2} = 0$$ $$y+\sqrt{-1(x^2-x)} = 0$$ ...
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2answers
88 views

If a,b,c are the 3 edges of a triangle then prove that $2<\frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b}-\frac{a^3+b^3+c^3}{abc}≤3$?

What I found, that is Since the sum of any 2 sides of a triangle is greater than the third if follows that $a + b > c$ or $\dfrac{a+b}{c} > 1$ and so ...
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1answer
30 views

Finding the count of paths with K turns from corner to corner in a square box

I'm having trouble understanding the solution given for the problem here: http://www.codechef.com/DEC11/problems/MOVES/ Given a square table sized $N \times N$ ($3 ≤ N ≤ 5000$; rows and columns ...
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1answer
38 views

Alternative Solution to a complex numbers problem

Let $z \in \mathbb C$, such that $z = x+ix, \; \forall x \in \mathbb R^* $ Prove that $$K(z) = \frac {z^4 + z^8 + \cdots+ z^{4n}} {iz^2 + i^5z^6 + \cdots+i^{4n-3}z^{4n-2} } = \mathrm {Im} ...
5
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1answer
83 views

Subset of $\mathbb{R}^4$ such that the intersection with a hyperplane is dense and does not contains $4$ coplanar points.

Does it exist a subset $S$ of $\mathbb{R}^4$ such that for all affine hyperplane $H\subset \mathbb{R}^4$, the set $H \cap S$ is dense in $H$ and does not contains $4$ coplanar points? More than ...
4
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1answer
36 views

Is a symmetrical fair odd-faced die posssible?

I can't come up with a scenario for a 5-faced die, but maybe 9 or 27 is possible?
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1answer
46 views

Brouwer theorem

Is the Brouwer's fixed point theorem true for the topological space '+' sign(cross)? $$ + = \left( [-1,1] \times \{0\} \right) \cup \left( \{0\} \times [-1,1] \right) $$ I have tried using spencer's ...
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1answer
29 views

Can Point inside quad be determined with angles alone?

Given A Quad($C$, $D$, $E$, $B$) and Points $A$, $G$, $F$ Question Is it possible by calculating the angles between points to determine whether a point is inside (including on), or outside the quad ...