Questions tagged [homothety]

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18 questions
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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 ...
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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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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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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??, ...
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Hartshorne - Geometry: Euclid and Beyond: Given two lines and a point, construct a circle passing through the point and tangent to both lines.

What is a proper solution for this problem? It has been bugging me for a good while.
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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 $(0,0)$...
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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 $(0,0)$...
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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 ...
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 $Q$.
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 ...