# 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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### Why am I getting the wrong formula for the area of a dodecagon?

More likely than not, I'm just making a simple algebraic mistake, but I can't seem to find it and so I would like some help. Divide a (regular) dodecagon into $12$ congruent isosceles triangles with ...
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### Prove transformation isn't a translation

I have these maps from the plane to itself where $X=(x,y)$: $f(X):=(y,-x)$ $g(X):=(x+2y,y)$ I need to compute $fg$ and $gf$ and show that none of these compositions are simply translations or that ...
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### Is it possible to think solid as the orthogonal projection from 4D?

Get an orthogonal projection's area from this fomular $S=S^{\prime}\cos{\theta}$ when the angle between two planes is $\theta$, is a popular idea, so I wonder is it possible to get a volume of various ...
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### Computing the set of fixed points of a map

Compute the set of fixed points of the following map: $f(X) := (y,-x)$ when $X=(x,y)$ So for this, do I just have to solve the system of equation such as: $x=y$ $-x=y$? Plugging the first into ...
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### Given two adjacent sides of a rectangle are equivalent, prove that the quadrilateral is a square.

In Geometry class today, we were talking about quadrilaterals and the types of them. I was wondering that if, given a rectangle with two adjacent equal congruent sides, if that was enough to prove it ...
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### I have a hard time understanding this simple theorem: “If two lines intersect, then exactly one plane contains the lines.”

I'm sorry if this is an extremely simple question, but I'm honestly having a hard time understanding a theorem in my geometry book. Here is the theorem: "If two lines intersect, then exactly one ...
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### UnFlattening a 1/2 Triaxial Ellipsoid: Reconstructing a Squashed Tortoise

BACKSTORY: I have a flat tortoise. I need to figure out its original dimensions. I'm a paleontologist, and the site I'm working at has produced a [Hespertestudo crassiscutata], a giant tortoise ...
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### Easiest way to prove a congruency

Let's take ABCD a convex quadrilateral, such that the diagonals AC and BD are perpendicular, and the angles ABC and CDA are equal. What is the easiest way to prove that the triangles ABC and CDA are ...
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### Is wolframalpha wrong (Plotting inequalities)

I just wanted to plot a simple inequality: $$-x \geq 4$$ and wolframalpha gives me the following plot: But I think it should look like this: Am I correct? If so why is wolframalpha producing such ...
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### Is the converse true too?

LDMJ is a circle centered at O. Point K, on DJ, bisects chord LM. DSJ is another circle drawn using DJ as diameter. If $\alpha = 90^0$, then KS = KL. This can be proved by applying “power of a ...
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### Missile Guidance Course Correction

Background: I am controlling a simulated (programming) missile in a 2D space (no drag). The missile knows which direction it wants to go (the intended velocity vector direction). It is always ...
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### Show that chordal metric is topologically equivalent to the Euclidean metric

Consider $$d(x,y)=\frac{2\|x-y\|}{(1+\|x\|^2)^{1/2}(1+\|y\|^2)^{1/2}},\hspace{5mm}x,y\in \mathbb{R}^n.$$ $d$ is a metric in $\mathbb{R}^n$ known as chordal metric. I want to show that this metric is ...
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### Determining whether the shape is a rectangle

While solving a problem, I came across a little hump which is impeding a pure solution. If there is a quadrilateral ABCD where $\angle B = 90^\circ$ and $AD = BC$ and $\angle D = \angle C$, is it ...
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### Bounds on the angles between four unit-length vectors in three dimensional Euclidean space

Let's consider four unit-length vectors $\mathbf{s}_i$, $i=1,2,3,4$, in three-dimensional Euclidean space. Let $\theta_{ij}$ be the angle between $\mathbf{s}_i$ and $\mathbf{s}_j$. Given the set of ...
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### Is this really an open problem? Maximizing angle between $n$ vectors

It is well known that the trigonal planar molecule (with bond angle $\alpha=120^{\circ}$) and the famous tetrahedral (with bond angle $\alpha\approx 109.5^{\circ}$) maximizes the angle between the ...
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### How to prove that a collection of epsilon balls generates a basis for topology?

Specific question: "Suppose $X$ is a three dimensional Euclidean space with the standard Euclidean metric. Let $Y$ be the subset defined by $Y=\{P_1$ s.t. $P_1=(a_1,b_1,c_1)$ and $c_1=0\}$ and use ...
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### Maximize distance to closest vertex inside triangle

Question: Let $\Delta ABC$ be a triangle. For any point $P$ inside or on the boundary of triangle, define $d(P)=\min\{\overline{PA},\overline{PB},\overline{PC}\}$. Find the maximum of $d(P)$ (in ...
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### Proving angles are supplementary in isosceles triangle

Let $ABC$ be a triangle with $AC=BC$, and let $P$ be a point inside $\triangle ABC$, satisfying $\angle PAB=\angle PBC$. If $M$ is the midpoint of $AB$, show that $\angle APM+\angle BPC=180^{\circ}$. ...
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### Median points collinear: At least one outside triangle

Suppose that ABC is a triangle and that $A'\in l_{BC}$, $B'\in l_{AC}$, and that $C'\in l_{AB}$. Prove that if $A', B', C'$ are collinear, then at least one of these points must be outside of the ...
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### Property of all k-dimesional shapes

A close friend showed me an intuitive pattern that given a k-dimensional convex polytope P, if one doubles every sidelength to generate a $P'$ it is possible to fit $2^k$ copies of $P$ within the ...
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### Concurrence of lines through triangle midpoints [duplicate]

Suppose that points $A, B, C$ form a triangle and that $A'$ $\in$ $l_{BC}$ and $B'$ $\in$ $l_{AC}$ and $C'$ $\in$ $l_{AB}$ are such that the lines $l_{AA'}$, $l_{BB'}$, $l_{CC'}$ are concurrent at G. ...
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### Concurrence through a triangle

Suppose that points A, B, C form a triangle and that A' $\in$ $l_{BC}$ and B' $\in$ $l_{AC}$ and C' $\in$ $l_{AB}$ are such that the lines $l_{AA'}$, $l_{BB'}$, $l_{CC'}$ are concurrent at G. ...
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### Two new conics associated with two triangles inscribed a conic

1-Let $ABC$ and $A'B'C'$ be two triangle inscribed a conic. Let $P$ be a point on the conic. $PA$ meets $B'C'$ at $A_1$, define $B_1, C_1$ cyclically. $PA'$ meets $BC$ at $A'_1$ define $B'_1, C'_1$ ...
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### $\pi$ is dependent on properties of geometry, assuming that we define it as $C/d$. Could then the $\pi$ also be an integer?

$\pi$ is dependent on properties of geometry, assuming that we define it as $C/d$. Could there be a geometry where $\pi$ is a rational number or an integer?
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### Prove that $u\cdot v = 1/4||u+v||^2 - 1/4||u-v||^2$ for all vectors $u$ and $v$ in $\mathbb{R}^n$

I need some help figuring out how to work through this problem. Prove that $u \cdot v = 1/4 ||u + v||^2 - 1/4||u - v||^2$ for all vectors $u$ and $v$ in $\mathbb{R}^n$. Sorry, forgot to ...
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### Common Intersection Point of ellipsoids

Suppose we have two ellipses in $2$-dimensions centered at the origin. It is easy to visualize that (unless one is contained in the other) they will have $4$ points of intersection. Is it known ...
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### Finding $y_{A,i}$ and $y_{B,i}$ in this geometric relationship problem.

Finding $y_{A,i}$ and $y_{B,i}$ in this geometric relationship problem. I'm an high-speed aerodynamics student. I am studying a sweptback wing like in the figure below (in green). Notice that I ...
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### Probably very basic Euclidean geometry; Why is the following expression valid for a point along a straight line?

I am looking at constructible points in abstract algebra, particularly in $\mathbb{C}$. Alongside a proof of a theorem, I came across this expression which I cannot work out how it's been derived. It ...
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### Four Spheres Intersect Along Circles: Prove That Circles' Planes Are Either $\parallel$ to The Same Line, Or Have a Common Point

Problem: Let $\,A,\,B,\,C,\,D\,$ be four distinct spheres in a space. Suppose the spheres $A$ and $B$ intersect along a circle which belongs to some plane $P$, the spheres $B$ and $C$ intersect ...
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### Finding closest vector for all rows in a matrix

I have two matrices 1. D ($m \times n$) and 2. C ($k \times n$). Typically, $m \approx 10^4, n \approx 100, k \approx 100$. For each row r in D, I need to find the index of the row in C that's ...
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### Partitioning a convex object without cutting existing convex subsets

$C$ is a convex object in the plane. $D_1,\dots,D_n$ are pairwise-disjoint convex subsets of $C$ such that $D_1\cup\dots\cup D_n \subsetneq C$, like this: Is it always possible to partition $C$ to ...
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### Size of a point. [duplicate]

I know this may sound too simple or maybe too absurd to discuss, but I am having a hard time visualizing a point in space! In Euclid's Elements a 'Point' is defined as Something which has no part. ...
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### A vector which is perpendicular to two vectors not in the same plane

Assume that I have two vectors $v_1, v_2$ which are not parallel and they don't lie the same plane. How to find a third vector $n$ perpendicular to $v_1$ and $v_2$? You could take the cross prosuct, ...
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### Conjugating rotation by another rotation

If $g ∈ \mathrm{SO}(3)$ is the rotation about axis $p$ by angle $α$, and $h$ is a rotation mapping $p$ to another line $q$, then $g$ conjugated by $h$ is the rotation about $q$ by the same angle $α$. ...