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

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Finding an angle on a triangle inscribed in a circle

How is angle $\angle OBW_2$ from the solution equal to $\theta$? The original problem only gives the measure of $\angle AOB$ as $2\theta$. Is there a theorem? or simple geometry I'm not using. ...
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0answers
27 views

All triangles are equilateral

So I was watching a video in which a man has "proven" that all triangles are equilateral. He also said that there is somewhere an error but I really cant find. So can someone show me it or give me a ...
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0answers
73 views

Is there a real problem to which $1$ radian is the answer?

I can't recall if I've ever seen any problem related to angles, in math or engineering books, that would result in an answer like $$\alpha=1 \ \ \text{radian}.$$ The answers to such questions, I ...
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43 views

Decorating eggs

I came across this question as I have been helping someone decorate styrofoam eggs by gluing wrapping paper onto it. The question is quite simple but I found myself stumped. Given an egg what is the ...
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14 views

From baby Hartshorne: In problem 5.19 why do the two perpendiculars cut of equal line segments

I am having some problems proving the following theorem: Given an angle AOB with vertex O and a point P inside the angle construct perpendiculars PA, PB, where P is the point within the angle where ...
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8 views

An intuitive affirmation about convex sets - normal at the boundary of a convex set

Let $\Omega_1 \subset \Omega_2$ two open bounded sets in $R^n $with $\Omega_i$, $i=1,2$ convex and with $\overline{\Omega_1} \subset \Omega_2 $. Suppose that $\partial \Omega_2$ is $C^1$. Now fix $y ...
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1answer
32 views

Splitting polygon in half. [on hold]

Let $P$ be a convex polygon in the plane. Prove that there is a vertical line which splits P onto two polygons of equal area. I tried to use intermediate value theorem with no luck.
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1answer
26 views

Calculating if an object is blocked from sight by another object

Is there an equation to determine if an object at altitude A can be seen at altitude B if there is an object between them at altitude C? Something to do with triangles I think... I know it has ...
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1answer
24 views

'3-point' curve

If you have a loop of string, a fixed point and a pencil, and stretch the string as much as possible, you draw a circle. With 2 fixed points you draw an ellipse. What do you draw with 3 fixed points?
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1answer
19 views

Find the sum of the squares of all sides and diagonals of a n-gon inscribed in a circle.

With a circle with radius r and center A, for any homogenous n-gon -- find the sum of the squares of all sides and diagonals of the n-gon inscribed within the circle. I believe the general rule for ...
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1answer
18 views

Find the area of the convex quadrilateral when you have the value of one diagonal and it's intersection point

ABCD is a convex quadrilateral and E is the intersection point of their diagonals if $DE=3$ and $BE=12$ find $\frac{ADC}{ABCD}$ I know the length of one diagonal so that's $BD=15$ and their now two ...
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Sequence of non-collinear integer points.

This is a question from a British Olympiad, I've completed the first 3 but this one had me rather stumped. Given two points $P$ and $Q$ with integer coordinates, we say that $P$ sees $Q$ if the ...
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3answers
624 views

Can Number Theory be visualized?

So I was thinking about a hard euclidean geometry problem, when it hit me just how much more difficult it would become without the aid of a diagram. This got me thinking: Wouldn't it be great if we ...
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1answer
16 views

Which Points are not Contained in the Line

The Circle $$x^2+y^2-4x=0$$ is cut by a line $AB$ at two points. If $A$,$B$ and two other points $C(1,0)$ and $D(0,1)$ are Concyclic, Then which of the Following points are not contained by the line. ...
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11 views

Transform Confocal Ellipsodal to Spherical Coordinates

I heard that someone published a paper showing that the confocal ellipsoidal coordinate system can transform into the spherical coordinates under special limit evaluations, however I was unable to ...
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0answers
33 views

Relation between circular continuity and elementary continuity

I read somewhere that in minimal geometry(incidence, betweenness and congruence axioms) the circular continuity If a circle has one point inside and one point outside another circle, then the two ...
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1answer
51 views

Is it possible to split a unit cube in such way?

Is it possible to split a three-dimensional unit cube into $42$ tetrahedrons of equal volume with nonoverlapping interiors? The two-dimesional case is discussed in "On Dividing a Square into ...
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1answer
16 views

Problem with similarity of triangles and median.

I have the next problem: In the next image, MN // AB, PN = NC, QM = 8, BM = 6 and MC = 9. Calculate PM. First I tried to find similarities in the triangles formed by the parallel sides, ABC and ...
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8 views

I am not able to construct a finite model for incidence axioms and axiom of parallels

Disclaimer: I had probably used some kind of "extended" set of incidence axioms, which also includes planes. Even though this is part of my homework, I seriously doubt this is what I was expected to ...
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1answer
20 views

Proving that $KL$ bisects $AJ$ in a triangle?

Let $ABC$ be an acute triangle, and its incircle touch the sides $AB$ and $AC$ at $K$ and $L$. Let $J$ be the incenter of $\triangle BCD$, where $D$ is a point on $AC$ such that $BD=AB$. Prove that ...
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0answers
31 views

Find length, width and height of a cuboid [on hold]

Consider a cuboid (that is, a rectangular box / rectangular parallelepiped) with the following properties: the area of its top face is $240$ cm$^2$ the area of its front face is $300$ cm$^2$ the ...
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0answers
19 views

Are 'similar' differentials equal? Specifics enclosed.

I'm trying to show that two infinitesimally small changes in an angle are actually equal, i.e. I want to say that $d\phi = d\phi '$, where the change in $\phi '$ is caused by a change in $\phi$. Here ...
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2answers
12 views

Question regarding vectors within a circle in an $x$- and $y$-plane.

In an $x$ and $y$ coordinate plane, with respect to the points $A, B$, and $C$ on a circle of radius $1$, find the minimum value of $\vec {AB} \cdot \vec {AC}$ So far, taking $O$ as the origin I've ...
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5answers
86 views

Proof for $\forall x \in [0, \frac{\pi}{2}]\quad \sin(x) \ge \frac{x}{2}$

What is the proof for $\forall x \in [0, \frac{\pi}{2}]\quad \sin(x) \ge \frac{x}{2}$ ? Assuming it is true.
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Blocks in a layer? $X$ layers of blocks in a triangle, which $Y$ being the total number of blocks… (w/o using triangular numbers)

I have $32$ layers of blocks in a triangle. However, for the sake of variation, let's call that $X$. I have $1000$ square blocks total. We'll call those $Y$. The first layer of the triangle has $1$ ...
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0answers
10 views

Figuring out the major axis and minor axis of a 3D ellipse

If a 3d ellipse S={(x,y,z)|(x^2)/(2t) + y^2 less than or equal to one, z=t, 1/2 less than or equal to t less than or equal to 1} The answer book gives the major axis to be 1, but shouldn't the major ...
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0answers
10 views

Parametrizing regions of complex plane

Let $\Omega=\mathbb{C}\setminus \lbrace t e^{it} \ \vert t \in \mathbb{R}_{\geq0} \rbrace$ I need to write $\Omega= \coprod_{i=0}^{\infty} R_i$ where each $R_i$ is the region bounded by from $t=2k ...
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1answer
12 views

What separates the dot product from the scalar projection?

Just a little problem with geometric intuition here (or perhaps I just haven't slept in far too long!). I know that the scalar projection of vectors $ \vec{u} $ and $ \vec{v} $ is defined as $ ...
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2answers
40 views

Defining the equation of an ellipse in the complex plane

Usually the equation for an ellipse in the complex plane is defined as $\lvert z-a\rvert + \lvert z-b\rvert = c$ where $c>\lvert a-b\rvert$. If we start with a real ellipse, can we define it in ...
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1answer
42 views

Find equation of ellipse given two tangent lines at given points and a point on ellipse

I'm attempting to generate an ellipse for a stair simulation game of mine, and the inputs are: A point on the ellipse The slope of the tangent line to the ellipse at that point Another point ...
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2answers
31 views

Gradient of a line using Pythagora's theorem

I've been trying to solve a question but I am having some difficulties. See the below diagram. I also know that the gradient of line c is twice as high as b i.e. c rises twice as fast as b. ...
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1answer
53 views

does the volume of a ball remain constant under deformation?

I'm a psychology student and was reading Piaget, he says that the volume of a sphere (ball of clay) remains constant if we deform the sphere into a roll for example, If you take the limit case of the ...
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3answers
37 views

Prove that if $l$ is a line in the classical euclidean plane, then there is a point $p$ that lies on $l$

Suppose that $\mathbb{P}$ is a Classical Euclidean Plane (satisfies all five of Euclid's postulates). Can you prove that if $L$ is a line in $\mathbb{P}$, then there is at least one point $p$ in ...
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1answer
32 views

show that two angles in a circumcircle are equal

We have the following circumcircle to ARP where PR = PQ are tangents to the smaller circle I need to show that the angle a = the angle b, which is equivalent to show that RP = AP', or show that the ...
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2answers
87 views

Euclidean Geometry challenge.

Can someone help me on this one? I have found that $\frac{1}{(x+1)^2}+1=\frac{1}{x^2}$, but I can't solve the fourth degree equation that comes with it. There must be a easier way!
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1answer
24 views

Explanation for relationship between hypotenuse segments and leg lengths?

|` | ` x | ` | ` c a | z /` | / ` y |_/ ` |/|_____` b I'm new to this SE, but I have an SO account, so hello! Can ...
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2answers
45 views

How to understand sinus?

In $\Delta PQR$ we have $\angle PQR=60^\circ$, $QR=4$ and $PR=a$. For which values of $a$ are there 0, 1 and 2 triangles matching the description? I think I'm supposed to use the law of sines, ...
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1answer
51 views

Construction of a triangle given some special points ($O,H,I$)

I'm a newbie in this site. I tried to search if this question was already answered but I'm not sure on how to do it. The problem is: given three distincts points $O,H,I$ namely the circumcenter, the ...
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0answers
16 views

Finding a plane where a vector lies (and generating its normal)

I have a vector in homogeneous coordinates $l = (a,b,1)$ and want to calculate a plane from it (for later doing intersections in 3D coordinates and calculate collision coordinates) with regards to the ...
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1answer
19 views

I don't understand this from my lecture note. What is described by $x=X(\xi)$? Why $\xi(t)$ is a curve since $\xi \in R^{m-1}$?

I don't understand this from my lecture note. What is described by $x=X(\xi)$? Why $\xi(t)$ is a curve since $\xi \in R^{m-1}$?
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0answers
38 views

Geometric proof concerning circles

From a point P outside a circle draw two tangents to the circle touching at points $A$ and $B$. draw a secant line intersecting the circle at points $C$ and $D$. choose point $Q$ on chord $CD$ such ...
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2answers
42 views

Triangle in parabola

I have a problem. In my triangle one vertex is in the vertex of the parabola and two others are in parabola. This is a isosceles triangle and I know one angle in this triangle : 120 grades. The ...
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12 views

Difference in (the concept of the distance between a line and a point) between Euclidean and non-Euclidean Geometry

What is the difference in (the concept of the distance between a line and a point on a graph) between Euclidean and non-Euclidean Geometry. Is this concept the same in all kinds of geometry??
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2answers
24 views

Can ellipse equation be transformed through one of its foci?

Can we transform ellipse equation to represent an ellipse transformed by tilting it through its focus such that its center point moves in circular manner and one of its focus stays at constant ...
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1answer
41 views

Weil divisors fail over singular varieties

Let be $k$ an algebraically closed field. We know that if $X$ is an irreducibile, normal variety, one can associate to every rational function $(f)\in k(X)^*$ a Weil principal divisor $$(f)=\sum_{Y} ...
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0answers
73 views

Crossed Ladders Problem

Two ladders, one 10 meters long and the other 8 meters [long], have been placed in a trench as indicated in the opposite figure. Their point of intersection, M, is 3 meters from the base of the ...
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0answers
12 views

Rational representation of conics

Currently I'm beginning my study of rational curves (Rational Bezier and NURBS) all books that I've read tell me that is "well known" that conics can't be represented by Bezier or even a B-Spline. ...
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2answers
26 views

How can I use Menelaus' theorem here (Simson line)?

Given 4 points on a circle A, B, C, and P. Draw the orthogonal projections of P onto triangle ABC and call them $P_1, P_2,P_3$. Show that $P_1, P_2,P_3$ are collinear. After drawing this out, I ...
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1answer
12 views

Proving that $2$-D parabolic coordinates are orthogonal

How can we prove that the parabolic coordinate system in two dimensions is orthogonal? I tried using the dot product, but don't know where to start or what basis vectors can be used in two dimensions. ...
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1answer
37 views

How would I use vectors for this geometry problem?

Consider a quadrilateral ABCD. K, L, M, N are the midpoints of the segments AB, BC, CD, DA respectively. O is the intersection point of LN, KM. Let P and Q be the middle points of the diagonals AC ...