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
24 views

A scalene triangle with no right nor obtuse angle

I want to find the area of the perfect triangle, i.e. a triangle with no particularity whatsoever : no side shall be equal to another, no right angle, no obtuse angle. So I gave myself a segment $[...
2
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1answer
30 views

On a constant associated to equilateral triangle, its generalization and the golden ratio.

I guess many of you are familiar with the result described as follows: If ABC is an equilateral triangle, and P is any point on the incircle of ΔABC, then AP² + BP² + CP² is constant. See link below. ...
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3answers
55 views

Check if an ellipse is within an other ellipse

I have an ellipse $E_1$ centered at $(h,k)$, with semi-major axis $r_x$, semi-minor axis $r_y$, both aligned with the Cartesian plane. How do I determine if another ellipse $E_2$ is within this given ...
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1answer
67 views

Is there a proof for this or we should accept that?

Why are two independent parameters necessary and enough for determining position of a point with respect to a reference point in a plane? In other words, I want to address a point from another ...
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0answers
8 views

Oblique projection onto orthogonal complement

I'm looking for an expression for the oblique projection along $B$ onto the orthogonal complement of $C$. Simply modifying the well-known oblique projection formula for the oblique projection anlong $...
1
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1answer
42 views

Let ABCD be a cyclic quadrilateral…

Let ABCD be a cyclic quadrilateral. Let r and s be the lines obtained by reflecting AB through the angle bisectors of $\angle CAD$ and $\angle CBD$, respectively. Let P be the intersection of r and s ...
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2answers
25 views

Let $T_1$ and $T_2$ be two circumferences…

Let $T_1$ and $T_2$ be two circumferences with centers $O_1$ and $O_2$ respectively, such that $T_1$ passes through $O_2$. Let $C$ be a point on $T_1$. Let $r_1$ and $r_2$ be the lines tangent to $T_2$...
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1answer
15 views

Unable to distinguish between intercepted arc and angle of triangle

$\triangle{PQR}$ is inscribed in circle C such that the measure of $\angle{PRQ}$'s intercepted arc is $70^o$ and m$\angle{PQR} = 50^o$. Find the measure of $\angle{QPR}$'s intercepted arc. When I ...
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0answers
50 views

Calculate lesser value that can take the side c=? [on hold]

EDIT: Consider a right triangle , it is satisfied that: $ab + bc + ac = 100$ Determine the smallest value that the side $c$ can take (without brute force)
3
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1answer
1k views

IMO 2016 Problem 3

Let $P = A_1 A_2 \cdots A_k$ be a convex polygon in the plane. The vertices $A_1, A_2, \ldots, A_k$ have integral coordinates and lie on a circle. Let $S$ be the area of $P$. An odd positive integer $...
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5answers
3k views

Why does $A^TA=I, \det A=1$ mean $A$ is a rotation matrix?

I know if $A^TA=I$, $A$ is an orthogonal matrix. Orthogonal matrices also contain two different types: if $\det A=1$, $A$ is a rotation matrix; if $\det A=-1$, $A$ is a reflection matrix. My question ...
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0answers
22 views

Geometry Of Unitary Transformations

Ever since I first took Linear Algebra, I have over time realized how concepts like determinants, eigenvalues, diagonalization, orthogonal transformations and so on have very intuitive geometric ...
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1answer
15 views

Ratio of bisected cevian in triangle given intersection point

I have the coordinates of points $A$ $B$ and $C$ that form triangle $\triangle ABC$, and the coordinates of a point $D$ inside of $\triangle ABC$. Imagine a cevian, connecting points $A$ and $D$, and ...
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1answer
22 views

How to find the corresponding 2D Cartesian coordinates from 3D ellipsoid?

I want to implement trilateration algorithm in short distance (maximum 50 meters), so instead of calculating intersections among spheres, I can roughly deal with it as finding intersections among ...
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1answer
26 views

Taylor's Theorem for $C^{\infty}$ functions

I am reading Tu's Introduction to Manifolds, the part where he derives a Taylor theorem (with remainder) for $C^{\infty}$ functions for points that belong in a set U that is star shaped with respect ...
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2answers
25 views

How many solutions exist for the intersections of these two line? Analytic Geometry

These are the questions I, just want to make sure i did the steps correctly 1.How many solutions exist for the intersections of these two line? (x,y)=(1,6)+s(3,-2) (x,y)=(4,4)+t(-6,4) 2.Find the ...
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1answer
55 views

If $a^2+b^2 = c^2$ and $a,b,c$ are sides of a triangle, can we say the triangle is a right triangle?

This may in fact be a silly question. Pythagoras tells us that $a,b,c$ are sides of a right triangle, then $a^2+b^2=c^2$. But is the converse true, that is if $a,b,c$ are sides of a triangle such ...
2
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1answer
57 views

How to calculate the ABV of an alcohol-infused watermelon?

I am trying to calculate the approximate ABV (alcohol by volume) of a watermelon that I am saturating with vodka. Details If I were to cut the watermelon in half length-wise, its face would be an ...
2
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2answers
56 views

Decomposition to rotation around arbitrary axis

In 3d, I have a $4\times4$ matrix $M$, which has only a rotation part and a translation part. In other words, I can compute $X'=RX+T$ ( with $R$ a $3\times3$ rotation matrix, $T$ a vector for the ...
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0answers
37 views

Three disks unique intersection contained in another circle

Let three disks:(assume a disk is the set of points of distance at most r from a center point) one centered at a with radius r_a, One centered at b with radius r_b, And one centered at c with radius ...
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1answer
50 views

I need someone to explain this shape to me.

The hexagon fits in the circle perfectly, the circle fits in the square perfectly. but the hexagon doesn't fit in the square perfectly. Doesn't this defy this formula below? a = b b = c a = c (what ...
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0answers
12 views

Getting starting/endings points Related to Displacement Vector

I am using this resource to calculate the distance between two 3d line segments. At the end it provides the 3D Vector dP. The length of this vector provides the correct distance between the two lines ...
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1answer
39 views

Distance between points on a circle [on hold]

A, B, C, D are points on a circle with AB=5 cm, BC=12 cm, AC=13 cm and AD=7cm. What is the closest approximation of CD?
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0answers
31 views

Iterating three tangent circles using Malfatti Circles

First, construct three tangent circles (blue circles), then construct the triangle joining their centers. Then construct three Malfatti Circles for this triangle (green circles). Go on. What I'm ...
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1answer
142 views

Israel tst 2011 geometrical inequality

Inside an equilateral triangle of area $S$ lies a point, whose distances to the vertices are $x,y, z$. Prove that $xy + yz + zx \geq \frac{4}{\sqrt{3}} S$ I haven't got any idea yet. But I guess ...
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4answers
5k views

How to determine if 2 points are on opposite sides of a line

How can I determine whether the 2 points $(a_x, a_y)$ and $(b_x, b_y)$ are on opposite sides of the line $(x_1,y_1)\to(x_2,y_2)$?
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2answers
34 views

How to rotate a line based dimensions of a piece of paper

I have a line where I know the start and end point on a piece of paper with the dimensions of 8 1/2 inches x 11 inches. the start point is 5.6 inches from the right of the paper and 4 inches down ...
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0answers
32 views

Comp Questions-Enumeration, Rates, Numbers, Geometry [on hold]

For each integer from 0 to 999, Michael wrote down the sum of its digits. What is the average of the numbers that Michael wrote down? It takes Jacob one and a half hours to paint the walls of a room ...
2
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4answers
33 views

How to know if a segment is completely included between two lines?

I have three segments (not necessarily parralel): blue $((ax1, ay1), (ax2, ay2))$ green$((bx1, by1), (bx2, by2))$ red $((cx1, cy1), (cx2, cy2))$ and a $margin$ value which is the width of the sky ...
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3answers
18 views

Cubic centimeters

Simple question which applies to chemistry in a measurement context as i am trying to understand centimeters cubed. If we calculate a box's volume. The width, length and height of a box are $15.3, 27....
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0answers
21 views

Covariance Matrix of Uniform Distribution Positive Definite

Suppose that $B$ is a Lebesgue measureable subset of $\mathbb{R}^d$. Let $U$ be the uniform distribution on $B$. Let $x \sim U$, and let $M = \mathbb{E}[xx^T]$. What conditions on $B$ guarantee that $...
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0answers
32 views

Get user position on the viewport

I'm creating a game, and I want to get the user x and y position on the viewport, but the problem is, I only have the world size and the viewport size and the user position based on the world. I made ...
0
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1answer
43 views

Why does $2\pi$ divided by the number of sides of a polygon give a regular polygon?

If I have $2\pi/n$, where $n$ is the number of sides of a polygon, does the answer give me a length of sides necessary for me to draw the polygon with those $n$ sides as a regular polygon inside the ...
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5answers
69 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
48 views

Let ABCD be a parallelogram and let M be a point…

Let ABCD be a parallelogram and let M be a point on side AB. Let P be the intersection of side BC with the parallel line to AC that passes through M. Let Q be the intersection of AC with the parallel ...
2
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3answers
131 views
+200

A trigonometric problem when calculating distance to the boundary of a convex hull

Suppose we have a sphere and a point outside of the sphere. We denote the point outside as $v$ and the origin of the sphere as $x$. The convex hull of the sphere and $v$ should be like an ice cream ...
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2answers
44 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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0answers
138 views

sqrt(bc) inversion problems

Can anyone explain to me what $\sqrt{bc}$ inversion is? A problem on that topic would be helpful too. I know the basis of geometric inversion and I'm searching for methods to solve Olympiad geometry ...
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0answers
14 views

Getting points Related to Displacement Vector

I am using this resource to calculate the distance between two 3d line segments. At the end it provides the 3D Vector dP. The length of this vector provides the correct distance between the two ...
2
votes
1answer
67 views

Volume of the ellipsoid $(x+2y)^2+(x-2y+z)^2+3z^2=1$

Find the volume of the ellipsoid $(x+2y)^2+(x-2y+z)^2+3z^2=1$, using integration. It is clear that this is not centered at the origin. So, how do I find the limits for an integral? Any suggestion ...
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0answers
23 views

What do you call the projected curve of a circle/ellipse on a cylinder?

What do you call the projected curve of a circle/ellipse on a cylinder? (The figure shows a circle projected on a cylindrical surface) See Figure (Circle Projected on a Cylinder) EDIT: If a name ...
0
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1answer
51 views

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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1answer
37 views

Is this a correct question?

Q. Find the shortest distance between two non-intersecting lines passing through the points whose position vectors are a and b are parallel to vectors c and d respectively. My confusion is : two non-...
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3answers
221 views

Finding area of sector inside an triangle

I have been asked this question from a junior and could not solve the question in a simple way. I am asking help on this platform. For a triangle $ABC$, Points $D, E$ on $AB$, where $AD:DE:EB=2:2:1$....
9
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1answer
66 views

Finite subgroup of $\text{SO}(3)$ acts on set of points on unit sphere in $\mathbb{R}^3$ which are fixed via some nontrivial rotation in $G$

Let $G$ be a finite nontrivial subgroup of $\text{SO}(3)$. Let $X$ be the set of points on the unit sphere in $\mathbb{R}^3$ which are fixed by some nontrivial rotation in $G$. I have two questions. ...
0
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1answer
30 views

Given a point, a vertex or covertex, and the center, how do you find the equation of the ellipse?

Say you have three points, P, V, and C. Point V is a vertex or covertex of the ellipse we're trying to find, point P is a random point on the ellipse, and point C is the center of the ellipse. For ...
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2answers
44 views

Find area of this circle

Given the circle O with perpendicular diameters and a chord, find the area of the circle if $EF = 8"$ and $DE = 20"$ inches.
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5answers
110 views

How to show that any rectangle in ellipse must be oriented parallel to axes?

A problem which is often given as an exercise for students learning about calculus and finding extrema, is to find maximal possible area of a rectangle inside an ellipse. Such question was asked, for ...
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2answers
380 views

Maximum number of vertices in intersection of triangle with box

Suppose we have a triangle and a box in 3D. The intersection of the triangle with the (solid) box will be a polygon with some number of vertices (possibly zero). The vertex count will vary according ...
23
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7answers
799 views

Product of cosines: $ \prod_{r=1}^{7} \cos \left(\frac{r\pi}{15}\right) $

Evaluate $$ \prod_{r=1}^{7} \cos \left({\dfrac{r\pi}{15}}\right) $$ I tried trigonometric identities of product of cosines, i.e, $$\cos\text{A}\cdot\cos\text{B} = \dfrac{1}{2}[ \cos(A+B)+\...