# Tagged Questions

geometry assuming the parallel postulate of Euclid: in a plane, given a line and a point not on that line, there is exactly one line parallel to the given line through the given point.

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### intersection of three planes different cases, algebraic and geometric explanations

http://www.vitutor.com/geometry/space/three_planes.html Could anyone help me to understand the following cases $1.$ CASE (2.1), why rank of co-ef matrix is $2$ and augmented matrix is $3$? I can ...
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### Find radius of a circle using stewart theorm

A circle C of radius 5 cm and two circles C1 and C2 of radius 3,2 respectively . C1C2 touch each other externally and both touch C internally . A circle C3 touch C1,C2 externally and touch C ...
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### Algebraic solution for the value of $x$.

I solved this problem the fifteen years ago without numerically solving equations of degree 4, I was happy in a substitution that I avoid directly attacking equations of degree 4. Today my nephew, ...
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### Can anyone give me a solution with analytic geometry or complex Numbers?

The problem is a imo's problem. Triangle ABC has circumcircle H and circumcenter O. A circle R with center A intersects the segment BC at points D and E, such that B, D, E, and C are all different and ...
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### Geometric interpretation of the geometric mean of two numbers

$a$ and $b$ are any two (positive) numbers. A geometric interpretation of the arithmetic mean and the harmonic mean of $a$ and $b$ are line segments parallel to the bases of a trapezoid of lengths $a$ ...
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### About the ratio of the areas of a convex pentagon and the inner pentagon made by the five diagonals

I've thought about the following question for a month, but I'm facing difficulty. Question : Letting $S{^\prime}$ be the area of the inner pentagon made by the five diagonals of a convex pentagon ...
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### What algebra is generated by $\mathrm{O}(2)$?

The unit complex numbers can be identified with the $2 \times 2$ special orthogonal matrices $\mathrm{SO}(2)$. The problem with $\mathrm{SO}(2)$, however, is that its not closed under $\mathbb{R}$-...
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### A problem on triangle and its perpendicular bisectors.

I'm trying to solve the following problem : "In △ABC, coordinates of $B$ are $(−3, 3)$. Equation of the perpendicular bisector of side $AB$ is $2x + y − 7 = 0$. Equation of the perpendicular ...
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### a solid geometry problem

In the following 3D figure, we know that $AE \bot EC, AD \bot BD$, how to prove that $|ED| < |BC|$ ?
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### Length of side of biggest square inscribed in a triangle

I have seen that the length of each side of the biggest square that can be inscribed in a right triangle is half the harmonic mean of the legs of the triangle. I have not seen a rigorous explanation ...
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### A space more fundamental than Euclidean space [closed]

Summary: The mathematical physicist Paolo Budinich attributes to Élie Cartan the statement that the geometry of pure spinors is "more elementary" or more "fundamental" than Euclidean geometry, which ...
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### Alternate form for the vector field $x^3\cdot\hat{x} + y^3\cdot\hat{y}$

Is there an alternate form for the vector field $x^3\cdot\hat{x} + y^3\cdot\hat{y}$ in which we can write all in function only of the radius $r=\sqrt{x^2 + y^2}$ ? Thank you
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### Trying to find an isomorphism

I'm trying to find an isomorphism from Z31 to itself that maps 13 to 28.I used the extended euclid algorithm to find that 12 is the inverse of 13, but that's as far as I got. What do I do next? How ...
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### Do the tangents of two circles define concentric circles?

Given two non-overlapping circles, $R_1$ and $R_2$. The radii of $R_1$ and $R_2$ may be different. The distance between the centers of $R_1$ and $R_2$ is defined as $x$. Draw the four tangents ...
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### How to plot $N$ points on the surface of a $D$-dimensional sphere roughly equidistant apart?

Let's say I have a $D$-dimensional sphere with a radius $R$. I want to plot $N$ number of points evenly distributed (equidistant apart from each other) on the surface of the sphere. It doesn't matter ...
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### How to find the area of the following isosceles triangle

I am stuck with the following problem : What is the area of an isosceles triangle whose equal sides are $20$ cm and the angle between them is $30^{\circ}$ ? It is a nineth standard problem and ...
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### Fisheye equidistant projection mapping to fisheye stereographic projection?

I have a set of images captured by a wide-angle (fisheye) lens camera, and the projection is linear-scaled (equidistant). I would like to remap from this projection to fisheye stereographic, which is ...
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### A generalization of the Sawayama lemma

Let $ABC$ be a triangle, let $D$ be a point on the line $BC$. The Thebault circle is a circle tangent $AD, BC$ and the circumcircle (yeallow circles in the following figure). I give a ...
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### The intuition behind choosing this length?

In Problem 1, RMO 2004 there is a particular choice of length which leads to the solution, the length being that of the tangent from the foot of the perpendicular to the circle. Just a rundown of ...
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### How much water would it take to fill a 1m^4 tesseract? Is it infinite? Do I need a 4D liquid?

Apologies, as I'm in no way a mathematically knowledgeable person. So this question may be proposed weirdly, or very simple. It's been evading my intuition for a while now, and I need a little help ...
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### Is there a simple way to decide if a hyperboloid is one-sheeted or two-sheeted, given the quadric equation?

Let us say that we have a quadric equation, whose solution set lies in $\mathbb{R}^3$, and you know it's a hyperboloid. Is there a way to analytically decide through a criterion if the hyperboloid is ...
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### Water filling problem in Blocks - Algebra Question

Consider a rectangular plot comprising $n\times m$ square cells on which $nm$ cement blocks of various heights are stored, one block per cell. The base of each block covers one cell completely, and ...
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### How to define a transported version of the simplex $X = \{x \in \mathbb{R}^n_{+}| \sum\limits_{i = 1}^n x_i = 1\}$?

Let the simplex in $\mathbb{R}^n$ be denoted as $$X = \{x \in \mathbb{R}^n_{+}| \sum\limits_{i = 1}^n x_i = 1\}$$ So it looks like: I want to take a point $\bar x \in X$ (red dot) And drag it to ...
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### Is there a theorem of intersecting chords in an ellipse?

I found a well known theorem that if $A,B, C$ and $D$ are on the circumference of a circle and $AB\cap CD=P$ then $AP\cdot BP=CP\cdot DP$ . Is there anything generalization of it to an ellipse? Maybe ...