# 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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### Are there some undiscovered/unproved theorems about Euclidean triangles?

For a particular case of a figure called simplex, a triangle is surprisingly complicated (in my opinion). As an illustration, see the list of triangle topics on Wikipedia, and the Triangle page. The ...
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### Does it have to be a right angle?

Say you have a circle $O$ and a point on the circle $P$. From P, we create 2 points $A$ and $B$ on the circle such that $PA=X$, $PB=Y$, and the 2 points are on different sides of $\overline{PO}$ (...
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### Solving this without the concept of similarity?

Any point $X$ is taken on the side $BC$ of $\Delta ABC$. Prove that $AX$ is bisected by the straight line joining the midpoints of $AB$ and $AC$. This problem is trivial when one uses the concept of ...
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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 ...
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### pack equilateral triangle

I'm working on a problem of inscribing equilateral triangle for some time now and it goes like this : using only a foot rule and a compasses , show a way of inscribing an equilateral triangle into ...
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### Proving an exercise from my High School Geometry Class

In my class we are learning geometry and the instructor gave us this problem: Let $ABC$ be a scalene triangle with orthocenter $H$, and let $W$ be a point on the side $BC$, lying strictly between $B$ ...
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### Prove on Incenter and mid point.

Let the incircle (with center $I$) of $\triangle{ABC}$ touch the side $BC$ at $X$, and let $A'$ be the midpoint of this side. Then prove that line $A'I$ (extended) bisects $AX$.
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### How do I convert $2=\sec(θ)$ into rectangular? [closed]

For my homework, I need to convert the polar equation of the "curve" $\sec\theta=2$ into rectangular form. I assume this must be an equation between $x$ and $y$, but the the answer according to my ...
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### Intuitive explanation of Pascal's Theorem

I am wondering why Pascal's Theorem, as well as other 'Euclidean' results in projective geometry like Brianchon's Theorem should be true for not only circles, but also conics in general. Is there ...
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### Would Euclid be satisfied by the construction of the 17-gon given by Gauss?

In our lecture on Algebra we were given the following exercice: Construct the regular 5-gon using straightedge and compass. (only using elementary geometric reasonig) If you construct the length ...
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### What is the relationship between average distance between 2 random points in a cube, with min and max?

I wrote a computer program to simulate the minimum, average, and maximum distance between $2$ random points in a unit length cube (such as $1$ cubic foot). The minimum possible distance is $0$ and ...
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### Solve using Butterfly Theorem.

Let $PT$ and $PB$ be two tangents to a circle, $AB$ the diameter through $B$, and $TH$ the perpendicular from $T$ to $AB$. Then prove that $AP$ bisects $TH$.
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### Prove between Simson line & Nine point circle.

Prove that the Simson lines of diametrically opposite points on the circumcircle are perpendicular to each other and meet on the nine-point circle. I proved the first part of the problem but not able ...
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### Have historians responded to Raju's critique?

C. K. Raju has made some outrageous criticisms of the traditional take on Euclid in particular and Western history in general. Yes he has a book published on the subject with an apparently respectable ...
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### Does a set of $n+1$ points that affinely span $\mathbb{R}^n$ lie on a unique $(n-1)$-sphere?

In $\mathbb{R}^2$ every three points that are not colinear lie on a unique circle. Does this generalize to higher dimensions in the following way: If $n+1$ element subset $S$ of $\mathbb{R}^n$ does ...
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### A straightedge and compass construction: $\left(G,I,Q_a\right)$

Construct $ABC$ with straightedge and compass, given $G,I,Q_a$. $G -$ the intersection point of medians; $I -$ the center inscribed circle; $Q_a -$ point of tangent inscribed circle to the side ...
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### Problem on circles, tangents and triangles

Let $c_1,c_2,c_3$ be three circles of unit radius touching each other externally. The common tangent to each pair of circles are drawn (and extended so that they intersect) and let the triangle formed ...
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### Efficient assignment of tetrahedron's chirality

Suppose we have a regular tetrahedron delimited by four points $A_{1}, A_{2}, A_{3}, A_{4}$. There are 24 permutations of vertices, but there are only two distinct terahedra that cannot be ...
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### Hyperbolic plane shrinking

A very small area of the hyperbolic plane looks more Euclidean as the curvature approachs 0. Any more evidence? Or reference would help? Thanks
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### The grey area is equal to the white area

Problem. Show that the sum of the areas of the white regions is equal to the sum of the areas of the grey regions. All the angles between consecutive chords are $45^\circ$. A solution (not totally ...
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### Piecewise linear curve where the closest vertex always belongs to closest edge

Take a piecewise linear curve $L$ in Euclidian space, i.e. a an ordered set of points $P$ sequentially connected by straight lines $l_{i}$, each defined by two points $p_i$ and $p_{i+1}$. Some such ...
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### I have a convex hull (generated from a library) in 3D. I only have the vertices. How do I compute the volume of the hull.

I have a library (quickhull in C++) that I am using to create a hull from a set of points. I am able to see the vertices of the hull but not the facets. I would like to compute the volume of the hull. ...
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### Proof - Elementar Geometry (parallelogram)

Prove that by connecting midpoints of adjacent sides of a quadrilateral we get a parallelogram. I'm having problems with this piece of work for some time so decided to ask for help here. Though I'm ...
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### Intersection over union optimization

Let $\mathcal R \simeq \mathbb R ^4$ be a set of all rectangles parametrized by $(x, y, w, h)$ -- coordinates of center and length of edges. How can I solve the following optimization problem: \begin{...