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

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Permutations of geometric structure

Sorry about the title, i don't know how to describe this problem. I tried counting my way through this problem but kept getting the wrong answer(which is 12, by the way). Is there a more systematic ...
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More generalization of the Sawayama lemma

Let $ABC$ be a triangle, $P$, $Q$ be two isogonal conjugate. $AP$, $AQ$ meets (ABC) at $D, E$ respectively. Two lines through $D, E$ meet (ABC) at $T, N$ and meet BC at $G, H$ respectively. Let $PG, ...
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31 views

How to solve the question related to geometry.

The question is : If $AB$ and $CD$ be two chords of a circle meets at $E$ then show that $\frac {AE} {CE} = \frac {DE} {BE}$. I don't find any clue to solve it.Please help me.Thank you in advance.
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An inequality in a triangle-a strengthened versions of the Erdős–Mordell inequality

Let $ABC$ be a triangle, let $I$ be the incenter of $ABC$. From $I$ and $ABC$, define $D$, $E$, and $F$ as the points where the angle bisectors of $\angle BIC$, $\angle CIA$, and $\angle AIB$ ...
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2answers
27 views

proving that triangles $ABC$, $A'B'C'$ are congruent

Given $AD$ is a median to $BC$ in triangle $ABC$, and $A'D'$ is a median to $B'C'$ in triangle $A'B'C'$, and $AD=A'D', AC=A'C', AB=A'B'$. How can i prove that triangles $ABC$, $A'B'C'$ are congruent? ...
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sectional curvature of hyperbolic space

For a pair of $(X,Y)$ of linearly independent vectors in $T_pM$, $p\in M$, the sectional curvature is defined as $$K_p(X,Y)= - \frac{<R(X,Y)X,Y>}{|X|^2 |Y|^2 - <X,Y>^2}$$ The problem I'm ...
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29 views

Given the angles between the edges of a triangular pyramid, what is the angle between any one edge and it's opposing side?

Rendition in paint You have a triangular pyramid with the edges a, b, and c. You know the angles between those edges - alpha (between a & c), beta (between b & c), and gamma (between a & ...
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2answers
25 views

Find the distance between the center of an unit circle and an internally tangent circle, which's tangents meet at the outer's center in a 72° angle.

There is a circle c, that has a center O, and a circle d that's internally tangent to it. If there are two tangents of d that meet at O, making a $72°$ angle, what's the shortest distance between the ...
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6answers
775 views

Proof that two non-parallel planes must intersect?

I managed to find, by enumeration, the intersection point of two planes $ax+by+cz+d=0$ and $ex+fy+gz+h=0$, in all possible cases (with the condition that the planes are not parallel). But this is a ...
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25 views

Overlapping area of two circle's crossing it's center i.e., length of overlapping is greater than r of the circle. Circle's has equal area.

Let there be two circular coasters of equal area (and negligible height). The purpose of is to find how far the two coasters need to be moved on top of each other such that the area of the overlapping ...
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Alternate form for the vector field $x^3\cdot\hat{x} + y^3\cdot\hat{y}$

Is there an alternate form for the vector field $x^3\cdot\hat{x} + y^3\cdot\hat{y}$ in which we can write all in function only of the radius $r=\sqrt{x^2 + y^2}$ ? Thank you
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4answers
58 views

Why the angle $\alpha$ in both triangles must be the same?

I've managed to understand the proof of the formula for the sum of cosines, but there is one detail which I couldn't uncover: In the following picture, Why the angle $\alpha$ in both triangles must be ...
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2answers
39 views

Does line-onto-line imply affine?

Let $n>1$ be an integer. Is every map $A : \mathbb{R}^n \to \mathbb{R}^n$ that maps lines onto lines (the image of a line is a line) affine?
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25 views

Swiveling a Rectangle or Cylander

I am looking for some concise way of describing a geometric transformation. For the purpose of this question I will refer to it as a swivel. To perform a swivel on some shape, you would make some kind ...
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1answer
27 views

Why is inradius $\times$ surface area equal to thrice the volume?

"Inradius" means radius of largest sphere that is tangent to all faces. For example: Cube - Surface area $= 6a^2$, Inradius $= a/2$, Volume $= a^3$. Sphere - Surface area $= 4\pi r^2$, Inradius $= ...
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What is this question asking for exactly?

So I'm given a question that asks: "A coffee filter is cone shaped with radius = 4 and height of 8. Suppose filter is filled with water up to a height of level h. Find an expression for the volume of ...
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Criterion for an affine isomorphism.

I am reading Don Taylor's book 'The Geometry of Classical Groups' and currently I am trying to understand the affine geometry section. There is a lemma which appears to be a criterion for a bijection ...
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2answers
36 views

Why can't I know if the figure is a rectangle, if angles c+d=180 and c=d?

I have a four sided figure, abcd (see the image, and ignore the EF part), where I know that angles c+d=180 and c=d. However, this isn't enough information to decide if this is a rectangle - why is ...
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8 views

when is a separable vector a product vector?

Consider a real tensor product space $V^{(1)}\otimes V^{(2)}$, and a set of vectors of the form $a\otimes b$. A "product vector" is defined as one that separates over the tensor product, e.g. $(a+b)\...
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1answer
57 views

Olympiad Books for Primary Students

I am a teacher of gifted program in primary school and currently I am developing Olympiad Curriculum (topic-wise) for my students. I have those topics that could need some help in terms of questions: ...
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3answers
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Problem including three circles which touch each other externally

The circles $C_{1},C_{2},C_{3}$ with radii $1,2,3$ respectively,touch each other externally. The centres of $C_{1}$ and $C_{2}$ lie on the x-axis ,while $C_{3}$ touches them from the top. Find the ...
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106 views

Books for maths olympiad

I want to prepare for the maths olympiad and I was wondering if you can recommend me some books about combinatorics, number theory and geometry at a beginner and intermediate level. I would appreciate ...
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How to estimate the maximum projection area of a set of spheres?

I have a set of spheres P. The spheres have a known, finite range of radii. It seems that there must be at least one 2 dimensional plane such that the bounding circle around the projection of P onto ...
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2answers
23 views

Calculating length of vertical line bisecting parallel arcs

I have 2 arcs, offset from one another (never intersecting) and a vertical line through them both (NOT at the center of the arcs). Is there a way to calculate the vertical distance between the 2 arcs? ...
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1answer
35 views

Integrating Over a Product of (Non-Separable) Piecewise Functions (Hyper-Solid Angle of a Convex Polyhedral Cone)

My problem is as follows: given a function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ where $n$ is some integer of order 10 and $f$ is defined by a product of (non-separable) linear piecewise functions, ...
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2answers
36 views

What would be the area of this Red Marked points? And how to calculate this?

I have been given the length $L$ and the width $W$ of a rectangle and the radius $R$ of circle which is situated in the center of the rectangle . I need to find the area of the red marked portion. ...
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0answers
49 views

Formula for the area of a trapezoid when only points are known

I am making a personal website and the main menu consists of scaled trapezoids. I can find the coordinates of the points for each trapezoid, and I know the parallel edges. I need a formula that can be ...
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2answers
56 views

Understanding Conics in Pencil

In a paper I'm reading about ellipses they talk a lot about "pencils of conics", after looking around on the web to learn more like this website: http://planetmath.org/pencilofconics I found some ...
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1answer
30 views

Geodesics using Euler-Lagrange

thanks for taking a look at my question. This is a homework problem from a section covering Euler-Lagrange equations. I'm asked to consider the arc length formula: $S = \int\limits_{{t_1}}^{{t_2}} {...
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2answers
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Finding area of a part of a circle

I have the values of $L$, $R$ and $W$ in the picture below. The circle is drawn though the center of the rectangle. And the circle will always intersect the rectangle. How can I find the area of the ...
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3answers
54 views

How to find the area of the following isosceles triangle

I am stuck with the following problem : What is the area of an isosceles triangle whose equal sides are $20$ cm and the angle between them is $30^{\circ}$ ? It is a nineth standard problem and ...
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Fisheye equidistant projection mapping to fisheye stereographic projection?

I have a set of images captured by a wide-angle (fisheye) lens camera, and the projection is linear-scaled (equidistant). I would like to remap from this projection to fisheye stereographic, which is ...
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Geometric Description Of a Set In The Complex Plane

$$S_1=\left\{z:Im\left(\frac{z-z_1}{z-z_2}\right)=0, z_1,z_2 \in \Bbb C\right\}$$ $$S_2=\left\{z:Re\left(\frac{z-z_1}{z-z_2}\right)=0, z_1,z_2 \in \Bbb C\right\}$$ Can someone help me with the ...
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40 views

Division of a square and value of a disk

I cam across this problem and I really don't know how to solve it. So you start with a square that has value 1. You divide this square in 4 so that each new square has a new value, as given by the ...
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1answer
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Find intersecting points on rectangle edges for line drawn inside it

Draw a rectangle ABCD. Draw a line inside it connecting any two edges GF. Draw a perpendicular bisector to line GF. At what points does the perpendicular bisector intersect the edges of the ...
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A generalization of the Sawayama lemma

Let $ABC$ be a triangle, let $D$ be a point on the line $BC$. The Thebault circle is a circle tangent $AD, BC$ and the circumcircle (yeallow circles in the following figure). I give a ...
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1answer
54 views

The function of distance between two points with time

Consider I have two points p and q, and a line segment l: y=mx+c (actually the enpoints of the segment are given). There is a circle with center q which is growing with time t, i.e. the radius r = k.t ...
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1answer
35 views

Shortest possible distance to locate an unknown road

You are stranded in the middle of a large desert and the only way home is a through a straight road, which unfortunately you do not know the location of. If the perpendicular distance from you to ...
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1answer
43 views

Prove that the Area of triangle whose vertices are $(0,0)$, $(b,a)$ and $(x,y)$ is $|by-ax|/2$

Prove that the Are of triangle whose vertices are $(0,0)$, $(b,a)$ and $(x,y)$ is $\displaystyle \frac{|by-ax|}{2}$. I found this problem in Number theory by George Andrews, but I wonder how it ...
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Intersection of Circles and Triangulation [on hold]

Tracking a Cellphone CT1 to CT2 = 700m CT2 to CT3 = 1200m CT1 to CT3 = 1350m Cell Phone is 600m from CT1, 650m from CT2, and 800m from CT3 Draw a circle in each Cell Tower, indicating the distance ...
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1answer
42 views

Show that three circles are coaxal

Let $A_1, A_2, A_3, A_4$ are collinear, $B_1, B_2, B_3, B_4$ are collinear. Such that $A_1, A_2, B_2, B_1$ lie on circle $(O_1)$, and $A_3, A_4, B_4, B_3$ lie on circle $(O_2)$. Let $MNPQ$ be the ...
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Proof of the Isoperimetric Theorem in Higher Dimensions

I have read a couple of nice proofs for the isoperimetric theorem in 2 dimensions. Is there a simple proof for the isoperimetric theorem in $n$ dimensions? In other words, how do you prove that the $n$...
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1answer
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Forming a expression of quadratic equation involving polygons

Six congruent isosceles triangles with equal sides $x$ cm are removed from the six corners of a paper in the shape of a regular hexagon of sides 20cm . The remaining portion is in the shoe of a 12 ...
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1answer
47 views

Calculating the intersection between two planes

EDIT: Let me give you the full problem at hand. We have an Orthonormal CS in E3*. According to that, we must find an analytical representation of a central projection c of E3* over the plane alpha: $...
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A nice problem of geometry (using inner product)

Problem Let $C$ be a sphere which is represented by $x^2 + y^2 + z^2 = 3 \quad (z\geq 0)$ and define two planes $\alpha, \beta$ as below \begin{align} \alpha &: 3x + \sqrt3 z + 3 = 0 \\ \...
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How to find the centroid of the intersecting region between three circles of differing diameters

This question is a follow-up to this question I asked earlier which deals with finding the midpoint of the intersecting region of two circles of differing diameters. Using the parametric equation of a ...
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Centroid of circle intersection

A region $R$ is bounded by the circle radius $4$ centred at $(5,0)$, the circle radius $2$ centred at $(0,0)$ and the x-axis, as shown. What is the centroid of this region? Finding the top ...
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Least area of maximal triangle inside convex $n$-gon.

A previous question Maximal area of triangle inside a convex polygon asks to show that for any convex polygon $P_s$ of $s$ sides having area $1$ there is a triangle contained in that polygon with area ...
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1answer
26 views

The intuition behind choosing this length?

In Problem 1, RMO 2004 there is a particular choice of length which leads to the solution, the length being that of the tangent from the foot of the perpendicular to the circle. Just a rundown of ...
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2answers
22 views

Sum of scores for square partition

A square is partitioned into some rectangles. For each rectangle with side lengths $a,b$, its score is $a^2+b^2$. It turns out that the sum of scores of all rectangles is equal to the score of the big ...