# 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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### Rectangle inscribed in a circular sector of angle 60

My apologies if this has been asked before. Given a circular sector, say of radius $r$, with internal angle $60^{\circ}$, construct a rectangle inscribed in that sector so that the length of the ...
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### Checking if vector crosses the simplex

Let assume that I have a point in $x \in \mathbb{R}^n$ Also I have a non-zero vector defined by it's endpoint attached to this point. The third thing I have is a simplex of $\dim=n$, such that the ...
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### Show that if a convex polygon is entirely inside another convex polygon, then the outer polygon has perimeter greater than or equal to the inner one.

Can anyone help me out with proving this statment? "Show that if a convex polygon is entirely inside another convex polygon, then the outer polygon has perimeter greater than or equal to the inner ...
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### Proof/justification that a circumscribed regular polygon has a perimeter greater than the circumference of the circle?

According to Archimedes, the perimeter of any circumscribed regular polygon is greater than the circumference of the circle. ie: http://www.themathpage.com/atrig/Trig_IMG/eval1.gif This does seem ...
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### Minimum Perimeter of a triangle

I have been playing the app Euclidea, I have been doing quite well but this one has me stumped. "Construct a triangle whose perimeter is the minimum possible whose vertices lie on two side of the ...
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### Defining perpendicular lines in the 3D space

Is there a universal agreement about the definition of "perpendicularity" between two stright lines in the 3 dimensional euclidean space? Do they need to meet or it is enough to have perpendicular ...
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### How can I find ratio between area of triangle and area of quadrilateral?

I have a parallelogram $ABCD$. $E$ is center of $AD$. $O$ is center of $AC$ and center of $DB$. $F$ is the intersection point between $CE$ and $DO$. Point $G$ is the intersection between $EO$ and $AF$....
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### Prove that if $A \perp B$ and $B \perp C$, then $A \parallel C$

Suppose that $A$, $B$ and $C$ are non-zero vectors in $\mathbb R^2$. Show that if $A$ and $B$ are orthogonal and $B$ and $C$ are orthogonal then $A$ and $C$ are parallel. I feel like this should be ...
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### What is the best way to inscribe a golden rectangle into a pentagon? Do more golden ratios emerge?

Below I drew a golden rectangle in a pentagon in Adobe Illustrator. What would be the best way to inscribe a golden rectangle into a pentagon as shown in the figure below in a mathematical manner? ...
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### General conic equation and coefficient matrices

For a general conic $Q(x,y)=ax^2+2hxy+by^2+2gx+2fy+c$ we define a matrix $A$ as follows: $A=\left( \begin{matrix} a& h& g\\ h& b& f\\ g& f& c\end{matrix} \right)$. Then we ...
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### Confusion on wording of an elementary geometry problem

I really want to know the following geometry problem is valid or not. (Please don't change the wording of the problem. Please answer it is valid or not. Please answer frankly.) "ABCD is a ...
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### Prove any line passes through at least two points

I've started reading Introduction to Algebra by Cameron, and I'm stuck on the first exercise. Q. Prove any line passes through at least two points using the axioms given below. Definitions: ...
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### How to find all those points whose distance from $x=(2,0)$ is minimum, using $\|x\|=|x_1| + |x_2|$?

The points must be in the closed ball $\{y : \|y\| \le 2\|x\|\}$. I know $|y_1|+|y_2|$ needs to be $\le 4.$ Other than that, I am confused about how to find all the points that are minimum distance ...
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