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Questions tagged [homothety]

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2
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1answer
53 views

$A=\lambda I_n\iff (\forall M,N\in M_n(\mathbb{R}),~ MN=A \Rightarrow ~ NM=A)$

here is my question: Let $n\geqslant 1$ and $A\in M_n(\mathbb{R})$. Show that $$\boxed{\exists \lambda\in\mathbb{R}^*,~ A=\lambda I_n\iff (\forall M,N\in M_n(\mathbb{R}),~ MN=A \Rightarrow ~ NM=A)}$...
3
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1answer
43 views

How gradient transform under homotety

Let $\Sigma^n \subseteq \mathbb{R}^{n+1}$ be a smooth hypersurface. Let $\lambda > 0$ be a constant and let define $\tilde{\Sigma} := \lambda \Sigma$. Let $f$ be a smooth function on $\Sigma$. ...
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0answers
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Homotheties in an affine Desarguesian space: conjugation and factors

Definition: a class of conjugated homotheties (conjugated by a translation) is called a factor. I really don't understand this definition of a factor. I believe it has something to do with the ...
0
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0answers
8 views

Proof. The set of homotheties with center $p$ forms a group, for each point $p$ in an affine space.

My attempt: Consider two non-trivial homotheties $\theta_1$ and $\theta_2$ with center $p$. A homothety is a dilation and dilations of an affine space form a group, meaning that $\theta_1 \theta_2^{-...
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0answers
14 views

R²/Plane Subset Equation With Plane Homothetic Transformation

Let's consider $H_k∶\ \left\{\begin{matrix}\mathbb{R}^2\rightarrow\mathbb{R}^2\\(x,y)\longmapsto(kx,ky)\\\end{matrix}\right.\ $. It is an homothetic transformation of $\mathbb{R}^2$ of center $(0,0)$...
3
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3answers
117 views

In $\triangle ABC$, $AD$ $\perp$ $BC$ and $GE$ is the extended line of $DG$ where $G$ is centroid. Prove that $GD$ = $\frac{EG}{2}$

Let $ABC$ be a triangle and in $\triangle ABC$, $AD$ $\perp$ $BC$ and three median lines intersect at point $G$ where $G$ is the centroid of $\triangle ABC$. The extension of $DG$ intersects the ...
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0answers
35 views

How to pick a “center” of a concave polygon?

I asked a question on how to scale concave polygons and a couple of people suggested some very clever solutions. The issue is that these solutions rely on picking an appropriate point $C$ in the ...
0
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0answers
20 views

Homothetic Transformation (Cycloid)

I could not proof the statement. For each point $(x,y)$, $x$ is not equal $0$, we can choose $r$ uniquely so that this point will lie on the first arch of the corresponding cycloid starting at $(0,0)$...
0
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0answers
19 views

Proving the Homothetic centers of two circles, are unique.

For any two circles there are two homothety centers, and I have to show this are the only possible centers of homothety . I started by supposing that a point Q is a center of homothety. Then, If we ...
-1
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1answer
24 views

Homotheties involved in a parallelogram

Given the parallelogram $ABCD$ in $\mathbb{A}^{2}$ and a point $P$ on the diagonal $AC$. Suppose that $AB \cap DP = \{Q \}$ and $BC \cap DP= \{R \}$. Show that $(D,P,R)=(Q,P,D)$. (For three collinear ...
1
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1answer
48 views

Why are circles homothetic in this problem?

I'm going through Solution 1 for problem G2 from IMO 2006 (https://www.imo-official.org/problems/IMO2006SL.pdf). In the penultimate paragraph they conclude that homothety $h$ takes circle $(ABP)$ to ...
2
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2answers
48 views

Show that this homothety takes $K$ to $M$

Let $C_1$ be a circle with centre $O$ and a chord $AB$,and consider a circle $C_2$ tangent internally to $C_1$ at $T$ and to $AB$ at $K$. Let $M$ denote the midpoint of the arc AB not containing $T$. ...
2
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1answer
79 views

Every conformal vector field on $\mathbb{R}^n$ is homothetic?

Consider $\mathbb{R}^n$ (for $n \ge 3$). Is it true that every conformal vector field on $\mathbb{R}^n$ is homothetic? A vector field is homothetic if its flow is a homothety- a conformal map with a ...
2
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3answers
81 views

Circle of Apollonius Textbook locus question

In the Locus 10, I get it till the point when it says that given triangles AOM and BOM, we have $$AM^2\ =\ OM^2+OA^2-2OA*OD$$And the same for $BM^2$, where does the author come to this conclusions??, ...
0
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4answers
55 views

I am having difficulty with the algebra related to a coordinate geometry question. [duplicate]

A line is drawn through the point $A=(1,2)$ to cut the line $2y = 3x-5$ in $P$ and the line $x+y = 12$ in $Q$. If $AQ$ = $2AP$, find the coordinates of $P$ and $Q$.
1
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1answer
99 views

Composition of homotheties does not look homothetic

Given an affine space $X$, let us define homotheties $h_1,h_2$ as $$ h_1(x) = c_1 + \lambda_1\overset\longrightarrow{c_1x} $$ $$ h_2(x) = c_2 + \lambda_2\overset\longrightarrow{c_2x} $$ for some $\...
2
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1answer
138 views
3
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2answers
501 views

Where is the homothety in the problem?

I have to solve the following problem using homothety but I don't see where it is. Given triangle $ABC$. $D$ is an arbitrary point inside the triangle. Points $M, E$ and $F$ are mid points of the ...