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

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0
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1answer
19 views

For any four distinct points $P_{1},P_{2},P_{3},P_{4}\in P^{1}(\mathbb{R})$ we have $(P_{1}P_{3};P_{2}P_{4})=1-(P_{1}P_{2};P_{3}P_{4})$.

Show that the cross-ratio has the following property: for any four distinct points $P_{1},P_{2},P_{3},P_{4}\in P^{1}(\mathbb{R})$ we have $(P_{1}P_{3};P_{2}P_{4})=1-(P_{1}P_{2};P_{3}P_{4})$. What is ...
7
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2answers
130 views

Right Triangle's Proof

A right triangle has all three sides integer lengths. One side has length 12. What are the possibilities for the lengths of the other two sides? Give a proof to show that you have found all ...
-1
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0answers
27 views

Geometry question: Hilberts parallel postulate and Proclus' attempt.

If all steps but one of an attempt to prove the parallel postulate are correct, then the flawed step yields another statement equivalent to Hilbert's parallel postulate. Assuming Aristotle's axiom, ...
0
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2answers
60 views

why soh cah toa is right?

i am confused by the sine of an angle, (it might appear evident for some of you but please i am not an expert ). sine of an angle is says to be the half of the magnitude of the chord of 2 time the ...
1
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1answer
30 views

Recalculating the radius

I have this steering scheme. Is it possible to calculate $r_o$ if I have $r_b$, $c$ and $b$?
2
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1answer
44 views

Why is $\mu=1$ clear?

I don't understand this reasoning from a solution. Q. Let $S^n=\{(x_0,...,x_n) \in \mathbb{R}^{n+1}:x_0^2+...+x_n^2=1\}$ and $X=S^n-\{1,0,0,...,0\}$ and $Y=\{(y_0,...,y_n):y_0=0\}$. Find $\mu$ ...
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2answers
68 views

Finding the maximum area of a triangle with a perimeter constrain

Using graphical methods, determine the dimensions of a right triangle that has the largest possible area, given that the perimeter cannot be larger than $P$. The final answer should be in terms of ...
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0answers
13 views

Triangle Medians Being Constructed Causes What To Happen ..?

The median of a triangle is a line segment joining the vertex of a triangle to the midpoint of the opposite side. What happens when all medians of one triangle are constructed?
2
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2answers
36 views

Can a line be a taxicab parabola?

For example, a line segment can be a taxicab ellipse if the sum of the distances equals the distance between the foci. So, can a line be a taxicab parabola?
0
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0answers
18 views

Prove that the point will go 3 times around ellipse

I'd like to prove that if a point $z$ goes once around ellipse with focus $2,-2$ then point $z^3-3z$ goes 3 times around some ellipse with the same focus. I was thinking (since ellipse is a set of ...
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0answers
9 views

Let S be a sequentially compact metric space, and let r > 0. Then S has a finite r-net.

Suppose $S$ has no finite r-net. We will define a sequence $(x_n)$ recursively, with $\rho(x_n,x_m) > r$ for all $m \ne n$. First, $S \ne \phi$ (since $\phi$ is a finite r-net in $\phi$)... I don't ...
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1answer
31 views

Can we use slopes in order to find the missing point in coordinate geometry?

Question: Plot the points $P(0, 3)$, $Q(2, 2)$, and $R(5, 3)$ on a coordinate plane. Where should the point $S$ be located so that the figure $PQRS$ is a parallelogram? Write a brief description of ...
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1answer
28 views

what are some great books on geometry(theory book) for imo? [closed]

Answers should contain at least 1 book on euclidean geometry, 1 book on coordinate geometry and 1 book on vectors. It is better if the books are available online to download
6
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1answer
80 views
+200

Locus of a point on a fixed-length segment whose endpoints slide along orthogonal lines

Suppose we have some segment $AB$ of constant length that slides in such a way that its endpoints are moving along orthogonal lines. Let $P$ be a point in the segment so that $|AP| = a$ and $|PB| = ...
3
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1answer
27 views

inequality between median length and perimeter

Is there an inequality between the sum of median lengths and the perimeter? If there is, can you specify a proof as well? I need to use this to solve a question. I tried using Apollonius theorem. ...
2
votes
1answer
51 views

Is the cuspidal cubic $\{y^2 = x^3\} \subset \Bbb R^2$ not smooth?

Cuspidal cubic $y^2=x^3$ in $\Bbb R^2$ "seems to be not smooth" intuitively because its pictured graph has a cusp at the origin. But I read from book that it is a smooth manifold. I feel so confused. ...
0
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2answers
46 views

Locus of centers of circles tangent to a given line and a given circle

I'm trying to find locus of centers of circles tangent to $y$ axis and tangent to the unit circle ($x^2 + y^2 = 1$). My try: Call $(x,y)$ the center of the circle. We know $$d( (x,y), ...
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2answers
40 views

Power of a point proof

I found the question on page 13 of this link. Let $P$ be a point inside a circle such that there exist three chords through $P$ of equal length. Prove that $P$ is the center of the circle. I ...
0
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1answer
21 views

Find point on rectangle where vector intercepts [closed]

I have a vector in the centre of a rectangle pointing out of the rectangle. The size of the rectangle is known. The vector is known. The magnitude of the vector is always greater than the distance to ...
0
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1answer
31 views

How to correctly find CSC of an angle?

Alright, so one of my questions is CSC (angle) -5. When I plug CSC in my calculator, it says "math error." I'm using a Casio fx-300 MS, and using shift + cos, then putting an angle, such as 90.
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3answers
123 views

Can a figure inside a circle be seen at right angle from any point on the circle?

A convex, closed figure lies inside a given circle. The figure is seen from every point of the circumference at a right angle (that is, the two rays drawn from the point and supporting the convex ...
2
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1answer
25 views

Is the surface of a torus 2-dimensional?

Unless I'm very mistaken, the surface of a torus is 2-dimensional, as is the surface of a sphere. The reason being that being on the surface you can only move in 2 dimensions, up or down is not well ...
0
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1answer
17 views

Which of the following quadratic forms defines a non-singular conic?

Which of the following quadratic forms defines a non-singular conic? (1). $x_{0}^{2}-2x_{0}x_{1}+4x_{0}x_{2}-8x_{1}^{2}+2x_{1}x_{2}+4x_{2}^{2}$. (2). $x_{0}^{2}-4x_{0}x_{1}+x_{1}^{2}-2x_{0}x_{2}$. ...
0
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1answer
16 views

Show that a projective transformation is unique.

Find the projective transformation $\tau \left ( \left [ 0,0,1 \right ] \right )=\left [ 0,1,0 \right ], \tau ([0,1,0])=[0,1,1],\tau ([1,0,0])=[1,1,1], \tau ([1,1,2])=[1,1,0]$. And show that such a ...
1
vote
2answers
51 views

What is a filled rectangle called, if anything?

In geometry, the set of points within a circle is called a disk (open disk if it excludes the boundary, closed disk if it includes it). Is there a similar notion for squares or rectangles? "A filled ...
22
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3answers
2k views

Where does the gap come from? [duplicate]

Can anyone tell me please where does the gap come from? Thanks and sorry if the question is not exactly relevant, I just didn't know where else to ask.
1
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1answer
42 views

How to graph in hyperbolic geometry?

I was given the following question regarding hyperbolic geometry: In the hyperbolic geometry in the upper half plane, construct two lines through the point $(3,1)$ that are parallel to the line ...
0
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2answers
25 views

Is it possible to decompose a triangle into quads without splitting edges?

By quads I mean four sided shapes. You can add vertex anywhere inside the triangle, but you can not add vertex onto existing edges, i.e., splitting them. I tried but currently it appears to be ...
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0answers
31 views

Finding a relation between three points in a small circle of a sphere

I have a relation as follows. I am given two points $C, D$ on a circle, and a point $P$ somewhere inside of it. I would like to find the quantity $|AC| |AD|$, where $A$ is a point on the circle lying ...
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0answers
27 views

How many different triangles have side lengths $x,y,z$ that satisfy $3x^3-yz^2 = z^3+4x^2-y$?

How many different triangles have side lengths $x,y,z$ that satisfy $3x^3-yz^2 = z^3+4x^2-y$? I was wondering about this and was wondering in general are there ways to solve such a question for ...
0
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0answers
16 views

How to determine what kind of curve in 3d geometry

I am having difficulty in determining type of given curve in 3d geometry.Is there any test in which I can differentiate between 1) Circle 2) Cone 3) Cylinder 4) Circle When equation of 3d curve ...
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3answers
58 views

A simple geometrical question regarding three circles and a line. Trigonometric construction. [closed]

In Figure 1 three tangential circles all have the radius of 1 or r. What is the ratio of the blue line to the yellow line in terms of r, and in terms of r=1?
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1answer
19 views

Isometries in $\mathbb{E}^2$

We define $\mathbb{E}^2 = (\mathbb{R}^2, d_E)$, where $d_E$ is the usual Euclidean metric on $\mathbb{R}^2$. We say that a function $f:X \rightarrow Y$, where $X, Y$ are metric spaces, is an isometry ...
1
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1answer
38 views

Coloring problem with equilateral triangles

Prove: If we color the plain with three different colours, then there will always be an equilateral triangle which has three vertices of the same colour. I have proved it for two colours but I just ...
2
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0answers
18 views
+50

Application of Jacobi's Theorem in Box Principle

Today I was going through Problem Solving Strategies by Arthur Engel, and found this in the chapter Box Principle Before the question it says it "treats a theorem of Jacobi and its applications" ...
0
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2answers
29 views

Do you need to measure and then prove that the distances are equal?

Question: My solution: Using the mid-point formula, we can easily prove that the coordinates of the point $M$ are $(\frac{a}{2}, \frac{b}{2})$. After this, we can use the distance formula to ...
1
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1answer
29 views

Parallelogram -diagonal-similarity problem

Given a parallelogram $ABCD$ . Points $M$ and $N$ are respectively the midpoints of $BC$ and $CD$ . Lengths $AM$ and $AN$ intersecting diagonal $BD$ consecutive points $P$ and $Q$. Prove that ...
2
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1answer
29 views

Partition a triangle into equal areas

A piece of wooden board in the shape of an isosceles right triangle, with sides $1$,$1$, $\sqrt{2}$ is to be sawn into two pieces. Find the length and location of the shortest straight cut which ...
1
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1answer
14 views

similar triangle problem in parallelogram with vertical lines

Can anyone help me with this task? I have no idea how to start. From the top $B$ of a parallelogram $ABCD$ lowered the vertical $BP$ and $BQ$ on the directions of $AD$ and $CD$ . From the top $D$ ...
0
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1answer
43 views

$ax^2+by^2+2gx+2fy+2hxy+c=0$ : Understanding the equation

Given any second degree equation in $x$ and $y$, $ax^2+by^2+2gx+2fy+2hxy+c=0$ is it possible to find out the centre and/or the axis of the conic section it represents? What information can I ...
4
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1answer
53 views

What is the radius of circle

There are two circles $C_1$ and $C_2$. The radius of the circle $C_1 = r$, and area of $C_1 = s$. The center of circle $C_2$ lies on the border of circle $C_1$. The area of the intersection of the two ...
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1answer
18 views

Avoid exponential growth when scaling

tl;dr; If I have to add a scale to a value which I expect to be X but is instead Y and the result, Z, is the most important part then how do I recalculate the scale to give result Z for Y instead of ...
0
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1answer
21 views

Let $K$ be midpoint of the hypotenuse of a right triangle $ABC$.On the leg $AB$ is a point $M$ s.t $BM=2MC$.Show that $MAB$ and $MKC$ are similar.

Let $K$ be the midpoint of the hypotenuse of a right triangle $\triangle ABC$. On the leg $BC$ is a point $M$ such that $ BM = 2MC$ . Prove that the triangles $\triangle MAB$ and $\triangle MKC$ ...
0
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1answer
53 views

Solving $\left(\;1-a\cos(\theta-\alpha)\;\right)\left(\;1-b\cos(\theta-\beta)\;\right)=\frac14\left(1-a^2\right)\left(1-b^2\right)$ for $\theta$

Let $0\lt a, b\lt 1$ be two constants. Then, how can I solve the trig equation ...
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2answers
19 views

Find all Points on the Surface at which the Tangent is Parallel to the Plane

The problem: Find all points on the surface $z=x^3+xy^2$ at which the tangent plane is parallel to the plane $2x+2y+z=0$ So I established $f(x,y,z)=x^3+xy^2-z$ and the normal vector determined from ...
3
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1answer
73 views

Show that the limit $\displaystyle \lim_{n\to \infty }\frac{a_{n}}{n}$ exists.

So $\left \{ a_{n} \right \}_{n\geq 1}$ is a sequence of real numbers and $C>0$ is a fixed constant. We assume that $a_{n+m}\leq a_{n}+a_{m}+C, \forall n, m\geq 1$. What is a good way to prove ...
6
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1answer
60 views
+200

The lattice points in the real cone of some semigroups are just the integer cone of that semigroup.

I'm trying to solve an exercise in Fulton's book on toric varieties, and have reduced it to the following: Let $M$ be a lattice of rank $n$ with $M \otimes \mathbb{R} = V$, and $S$ be a finitely ...
0
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1answer
16 views

Show that $T$ does not have any fixed points in $\mathbb{R}^{2}$ if and only if $T(P)=P+T(0)$, $\forall P\in \mathbb{R}^{2}$.

So $T:\mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ is an orientation-preserving rigid motion with $T(0)\neq 0$. What is a good way to prove this? Thanks a lot.
0
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2answers
48 views

Find the fourth missing coordinate of a square in a Cartesian plane.

Question: Plot the points $P(5, 1)$, $Q(0, 6)$, and $R(-1, 1)$ on a coordinate plane. Where must the point $S$ be located so that the quadrilateral $PQRS$ is a square? Find the area of this square. ...
0
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1answer
20 views

Distance of projection on a curve

Suppose there is a circle. Maybe, to be able to plot it in wxMaxima or Octave, $f=\sqrt{1-t^2}$. Then there's a second circle (or half-circle), $g=k\sqrt{1-(\frac{t}{k})^2}$, $k>1$ (e.g. 1.2). Now ...