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

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Prove that the circumcenter of $\triangle PIQ$ is on the hypotenuse $AC$.

In a right $\triangle ABC$ right angled at $B,BD$ is the perpendicular on $AC.P,Q,I$ are the incenters of $\triangle s ABD,CBD$ and $ABC$ respectively.Prove that the circumcenter of $\triangle PIQ$ is ...
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13 views

How to solve this task about circles and lines intercepting each other?

We have drawn some lines and circles on a paper. Every two has an interception, but none three goes through the same point. How many lines and circles have we drawn if we have 75 interceptions?
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2answers
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How can I see that the space curve given by $\gamma(s) = (\frac 4 5 \cos s, 1 - \sin s, - \frac 3 5 \cos s)$ forms a circle in space ?

Let $\gamma: \mathbb R \rightarrow \mathbb R^3$ be a space curve given by $\gamma(s) = (\frac 4 5 \cos s, 1 - \sin s, - \frac 3 5 \cos s)$. How do I see that $\gamma$ has image in $\mathbb R^3$ that ...
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1answer
18 views

How to prove $A( v \times w)$ = $\det A\cdot(Av \times Aw)$ , where $A\in \text {Mat}_{3 \times 3}(\mathbb R)$ is an orthogonal matrix.

How to prove $A( v \times w)$ = $\det A\cdot(Av \times Aw)$, where $A\in \text {Mat}_{3 \times 3}(\mathbb R)$ is an orthogonal matrix ($\det A = \pm 1$) and $v,w\in \mathbb R^3$. I've tried writing ...
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2answers
34 views

How to connect a line between 4 randomly placed points on a plane such that the line does not cross itself

You get 4 coordinates of points on a plain. You need to connect them all with a line. The line must not cross itself. What's your strategy?
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0answers
13 views

integral cohomology of real grassmannians

I have obtained that the cohomology rings $$ H^*(G_k(\mathbb{R}^\infty);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k]. $$ Also $$ H^*(G_k(\mathbb{R}^m);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k]/(\bar ...
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1answer
12 views

Area of a square inscribed in a circle of radius r, if area of the square inscribed in the semicircle is given.

If a square is inscribed in a semicircle of radius r and the square has an area of 8 square units, find the area of a square inscribed in a circle of radius r. I started by assuming that the side of ...
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2answers
21 views

Find the length of the chord given that the circle's diameter and the subtended angle

A chord of a circle subtends an angle of 89 degrees at its centre. Find the length of the chord given that the circle's diameter is 11.4 cm. The problem I have here is that I can't visualise this ...
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0answers
27 views

Find L for $r = \cos 3 \theta$.

Pictured above is the graph of $r = \cos 3 \theta$ for $0 \le \theta \le L$. Find the smallest value of $L$ that still produces the entire graph of $r = \cos 3 \theta$. I am having trouble starting ...
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1answer
25 views

What can we say about the areas of these two triangles?

Given triangle ABC. Let X be a point on AB, Y be a point on BC and Z be a point on AC. Now suppose we reflect X, Y, Z around the midpoint of the sides they are on and label the images X', Y' and Z'. ...
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1answer
41 views

Finding distance between the unit ball in $\mathbb{R}^2$ and the point $(1,1)$

Given the euclidian metric. $d(x,y) = ((x_1-y_1)^2 + (x_2 -y_2)^2)^{1/2}$ find the distance between the point $(1,1)$ and the set $A = \{x=(x_1,x_2) \in \mathbb{R}^2 : x_1^2 +x_2^2 \leq 1 \}$ Where ...
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1answer
25 views

Geometry: Perpendicular tangent

I came up with this but I have not been able to solve it. I would really appreciate any help. Let $ABC$ be a triangle and let $\omega$ be its circumcircle. Produce the internal angle bisector of ...
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0answers
14 views

How to find if a point coords is inside rhombus coords [duplicate]

A = 282, 34 (x,y) B = 59, 198 (x,y) C = 282, 359 (x,y) D = 509, 198 (x,y) E = 10, 10 So I have this image and I am trying to search if coords of point E are ...
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0answers
26 views

Calculate the diameter of the unit ball in $\mathbb{R}^3$ using the Euclidean metric.

So the question states, Let $B = \{x = (x_1,x_2,x_3) \in \mathbb{R}^3: x_1^2 +x_2^2 +x_3^2 \leq 1 \}$ be the unit ball in $\mathbb{R}^3$. Compute the diameter of $B$ for each of the following metrics. ...
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1answer
25 views

Equation of a parabola: Translations and rotation

I've tried to solve this problem: Find an equation of the parabola with vertex at point $(1,1)$ whose directrix is the line $x-2y=6$. It has to be solved using translation and rotation (coordinate ...
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1answer
50 views

Finding an angle on a triangle inscribed in a circle

EDIT: It appears as though the text has made a typo. The angle should be $\theta - \phi$. !! How is angle $\angle OBW_2$ calculated in terms of $\theta$ or $\phi$ if the only angle measures given are ...
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0answers
32 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
86 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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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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0answers
19 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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0answers
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
37 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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2answers
38 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
27 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
19 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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0answers
18 views

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
715 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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0answers
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
41 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
55 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
17 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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1answer
16 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
22 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
32 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
20 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
13 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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0answers
19 views

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
11 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
42 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
43 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
32 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
55 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
33 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
88 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!