# Tagged Questions

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

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### Changing the manifold, preserving the discrete spectrum

On a Riemannian manifold $M$, the Laplace operator $L$ is uniquely defined. If $M$ is not compact, then $L$ admits a continuous spectrum. Is there a way of "changing" $M$ and/or $L$ in say $M'$ ...
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### some notions on algebraic curve

1) I want to learn about algebraic curves and i'm confused, please correct me if i'm wrong : when we say an Affine algebraic curve over the field $F$ : here affine to distinguish it from projective ...
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### How can I count waypoints between a curve?

I have curve that is drawn between point A and B. I want to divide this curve to 100 smaller waypoints. How can I determine what these 100 waypoints are as coordinates, when I only know points A and ...
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### Decomposition of any point in the unit hypercube as a positive linear combination of polynomial number of vertices

I have function values at each of the vertices of the hyper cube. What would be a natural interpolation of the function to each point on and inside the cube that can be written as a positive linear ...
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### Intersection of Sphere and Line in $\mathbb{R}^n$?

This seems to me as a very simple and basic question, though I'm having trouble with it. The Problem Given a sphere $K\in\mathbb{R}^n$ with radius $r\in\mathbb{R}$ and center ...
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### Finding the radius of the largest sphere possible between a corner and another sphere

In a 3 dimensional Cartesian plane there is a sphere A that is in the first octant and is tangent to all coordinate planes. Now, imagine we want to find the another sphere B also tangent to all ...
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### Proving Stewart's theorem without trig

Stewart's theorem states that in the triangle shown below, $$b^2 m + c^2 n = a (d^2 + mn).$$ Is there any good way to prove this without using any trigonometry? Every proof I can find uses the ...
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### Coordinates of point on a line defined by two other points with a known distance from one of them

I have two points in 3D space; let's call them $A=(a_x, a_y, a_z)$ and $B=(b_x, b_y, b_z).$ Now, I need to place a third point, let's call it $C=(c_x, c_y, c_z)$, which lies on the line between $A$ ...
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### Maximum size of a rotated-then-cropped rectangle

With regard to topic/question New size of a rotated-then-cropped rectangle: The answer by Isaac, the maximum area is $b^2\csc\alpha\sec\alpha$ when $x=0.5b\csc\alpha = 0.5b/\sin\alpha$ seems to ...
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### How can I learn de casteljau algorithm? (from calculus)

I'm an highschool graduate who is currently waiting for college. Meanwhile, I'm trying to do a little project by myself. (Computer stuff) And yesterday, I found that I needed to deal with something ...
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### Two tetrahedra are congruent given a certain condition

This question is inspired by a Miklos Schweitzer problem, namely Problem 9./2007 Let $A$ and $B$ be two triangles in the plane such that the interior of both triangles contains the origin, and for ...
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### Name of this convex polyhedron?

Does anyone recognize / know the name of the convex polyhedron depicted below as the intersection of a Cuboctahedron and a Rhombicdodecahedron? Please note you have to interpret this picture and ...
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### Find the centre of a circle passing through a known point and tangential to two known lines

I am trying to find the centre and radius of a circle passing through a known point, and that is also tangential to two known lines. The only knowns are: Line 1 (x1,y1) (x2,y2) Line 2 (x3,y3) ...
In an old IMC Shortlist, I found the following problem: Given a triangle $T$, consider the equilateral triangles $T_1\subset T\subset T_2$ such that $T_1$ is the greatest equilateral triangle ...