Questions tagged [euclidean-geometry]
For questions on geometry assuming Euclid's parallel postulate.
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Are there four dimensional generalizations of the Reuleaux triangle and other solids of constant width?
Is there a four dimensional generalization of the Reuleaux triangle? What is it called, and what properties does it have?
Thank you!
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Solution requested for geometry puzzle [duplicate]
In $\triangle ABC$, $\angle BAC=2\angle ABC$ and $0^o<\angle BAC<120^o$. A point $M$ is chosen inside $\triangle ABC$ such that $BA=BM$ and $MA=MC$. Find the value of $\angle BCA$.
I have tried ...
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What is a gyrational square in this context?
This is a diagram of quadrilaterals and their duals. Within this diagram, there is a square (shown with 8 lines of symmetry) and right below that there is a "gyrational square" shown with no ...
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Prove that $\overrightarrow{AB}\cap \overrightarrow{BA} = \{A,B\}\cup \{X: A-X-B\}$
As far as I can tell, both of these are perfectly sufficient definitions for a line segment $\overline{AB}$. The book I am using defines them the latter way, but is the first definition also ...
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Why is the line/plane separation postulate necessary?
Usually when we introduce postulates it is so they are of some use to us. I do not see the reason for these two.
The Line Separation Postulate: Each point on a given line divides the line into three ...
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Hess' Proof of Fuss' Formula
I am trying to understand the short proof of Fuss' Formula in the paper "Bicentric Quadrilaterals through Inversion" by Albrecht Hess which is available here:
https://forumgeom.fau.edu/...
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Proving the relation between sides and diagonals of parallelogram without trigonometry and Pythagoras theorem
I am looking for a way to prove that the sum of the squares of the diagonals of parallelogram is equal to the sum of the squares of its sides. However, I would like to know whether it is possible to ...
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How to prove that a figure tiles the plane (for example certain pentagons)
Is there a general way(s) to approach proving that a certain shape tiles the plane? For example the type 4 pentagonal tiling found here:
https://en.wikipedia.org/wiki/Pentagonal_tiling
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Given a Set Of Points, Find a Set of Direction Vectors Such That All Resultant Lines Intersect at a Single Point
Given a set of $n$ points in 2D, how do I express a system of equations, where the unknowns are $n$ corresponding direction vectors, such that the lines defined by each point, direction vector pair $(...
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It is impossible to find a multi-surface whose surroundings are equal to its area and its size
I had a guess from three years ago that I couldn't prove
It is impossible to find a polyhedron whose perimeter (represented by the sum of the lengths of its edges) is equal to its area (represented by ...
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Proving two triangles congruent given two congruent sides and a congruent median
The title was a bit too short for me to fit the full details, so here's the scenario I have.
Prove that two triangles are congruent if in two triangles, the median from the common vertex and two sides ...
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Question about trapezoid [closed]
Good time of day! I have the following question.
Let $ABCD$ is a trapezoid with the base $AD$ and $\angle BAD + ∠ADC \neq 120^{\circ}$. Points
$A′$ and $B′$ are located symmetrical to points $A$ and $...
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A "New" Special Point in a Triangle.
I was playing with the software Geometry Expressions and I was exploring generalizations of special points in triangles (centroid, orthocenters, etc.) when I stumbled upon this construction.
J is ...
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Angle chasing problem with quadrilateral - possible approaches?
CONTEXT
I was recently working on some Langley-style problems, and wanted to construct some others, based on the "reverse engineering" approach I developed to solve them.
PROBLEM
The ...
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Relation between AD, BD and BC
In a triangle $ABC$, $\sphericalangle BAC = 100°$ , $AB=AC$. A point $D$ is chosen on the side $AC$ such that $\sphericalangle ABD = \sphericalangle CBD$, prove that $AD+DB=BC$.
I tried solving this ...
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Is the area of a triangle is less than that of its mean triangle of equal area?
Definitions: We generate random triangles using various distribution (uniform, gaussian etc) for $\theta$ to get the polar coordinates of the vertices $𝑟\cos \theta,𝑟\sin \theta$ in a circle of ...
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A problem about perpendicular lines and areas bounded by coordinate axes
I did not want to publish this problem and I wanted to solve it myself, but I was tired of dealing with this difficult problem. This problem, in a way, tells us that we need some theorems that start ...
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Can the Miquel Point lie on the Triangle?
Here is a link to the Wikipedia article regarding the Miquel point Theorem. https://en.m.wikipedia.org/wiki/Miquel%27s_theorem#:~:text=Miquel's%20theorem%20is%20a%20result,points%20on%20its%20adjacent%...
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What is the minimum and the maximum perimeter of a triangle with area $x$ that can be inscribed in a circle?
The area of the largest triangle that can be inscribed in a circle of raidus $1$ is $\displaystyle \frac{3 \sqrt{3}}{4}$ for a equilateral triangle and it also gives the perimeter is $3\sqrt{3}$. For ...
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Does the mean area of triangles with equal perimeter $p$ and circumradius $R$ have a local minima at $p=4R$?
Definition: Isoperimetric triangles are triangles which have the same perimeter and the same circumradius.
Isoperimetric area curve: The largest perimeter of a triangle that can be inscribed in a ...
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Approximate quadratic Bezier by a $1.0-\sqrt{x^2 + y^2}$ distance
Summary:
I have a triangle with points at $A=(0, 100); B=(100, 0); C=(100, 100)$ each point also has an $F$ value, this value is linearly interpolated between every other point and used to get the ...
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Distance between taking the shortest paths vs taking the longest paths
Assume we have a finite set of points A in $\mathbb{R^2}$ or $\mathbb{R^3}$. Is it true that if we start from a fixed point and travel to the closest unvisited point, the sum of these distances will ...
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How does Euclid's Fifth postulate not hold here?
If look at the Latitude lines - blue circles below - and pick one, each point $P$ not in one selected circle/line, will only lie in one other blue circle, circle here taken as line, and will follow ...
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How to enumerate unique lattice polygons for a given area using Pick's Theorem?
Pick's Theorem
Suppose that a polygon has integer coordinates for all of its vertices. Let $i$ be the number of integer points interior to the polygon, and let $b$ be the number of integer points on ...
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If $P$ is a point on a rectangular hyperbola with center $C$ and foci $S$ and $S'$, then $SP \times S'P=(CP)^2$
Is there a geometric proof to the statement below?
If $P$ is a point on a rectangular hyperbola with center $C$ and foci $S$ and $S'$, then $SP \times S'P=(CP)^2$.
I am staring at the triangle $\...
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Which parameters can we choose in order to solve this triangle construction issue
This is follow-on of a question asked yesterday, with real work shown under the form of sketches but not understandable. Visibly, the asker isn't used to formulate mathematics with sentences (his/her ...
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Help me make the sides and faces for some trapezohedrons (like a dice D10 or D14) with a circumsphere for EVERY desire
The goal: Make the face for a 10-sided dice or a heptagonal trapezohedron with a circumsphere, using as input one side of a face or the radius of the sphere.
Let us be on the same page:
Circumsphere: ...
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The addition of randomly directed straight line "slope" vectors in a 3-D Euclidean cartesian space
As I was trained in engineering and physics I sometimes get confused when trying to abstract my thoughts mathematically and this maybe a terminology problem, temporary misconception, for me as much as ...
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Conditions for equidistant points in space
Given a set of n distinct points in space, where n is a natural number greater than 1, what is the geometric structure that results from connecting each point to every other point with straight lines ...
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Finding the angle $x$ in $\triangle ABC$
What is the angle $x$ in the following diagram.
I could solve this problem by applying the Sine Rule for $\triangle ABD$ and $\triangle BDC$. After some simplification I got
$$\sin^2 x =\sin(45-x)\...
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Finding minimum value of $AX+XB-XX'$
Question: There are two points on a plane: $A$ and $B$. Find point $X$ such that value of expression $AX+XB-XX'$ is minimal where $X'$ is orthonormal projection of $X$ on $AB$
Tip to task is to draw ...
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Prove that the longest chord in an ellipse is its major axis.
It is easy to prove this by taking point $(a\cos\theta, b\sin\theta)$ on $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$. But I am looking for purely geometric proof without using trigonometry or ...
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Big picture: Geodesics on a spherical surface embedded in $\Bbb R^3$
Consider the closed surface $M$ embedded in $\Bbb R^3$ with $g:=ds^2=dx^2+dy^2+dz^2$ and with $M:=\log^2 x+ \log^2 y +\log^2 z=1.$
Then restrict the metric to $M$. Here is a 3D plot of $M$ embedded in ...
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angle Ceva's theorem
In triangle $ABC$, $P$ is point such that $\angle PAB = 42^{\circ}$, $\angle PBA = 54^{\circ}$, $\angle PAC = 6^{\circ}$, $\angle PBC = 12^{\circ}$. Find $ \angle PCB$.
I found the $ \angle PCB ...
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Finding the angle EDB in triangle ABC, where E is the intersection of the angle bisector of C with side AB and D is a point on BC
This was a question I encountered while looking at some weekly math questions my school had hung in front of the department last week: I was unable to solve it, and now that some time has passed, I'd ...
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Prove that certain quadrilateral is cyclic [duplicate]
Let $ABC$ be an arbitrary triangle. Let $D$ be an intersection of perpendicular bisector of side $AB$ and angle bisector of angle $ACB$ ($\angle ACD=\angle BCD$). Prove that quadrilateral ACBD is ...
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Seeking Cartesian to Hyperspherical Coordinates Conversion Examples in Higher Dimensions
I've recently coded a function that transforms Cartesian points into hyperspherical points. While I've verified its accuracy for 2D and 3D points, I'm finding it challenging to test for higher ...
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How to calculate the area of a truncated ellipse?
I’m in a bit of an oddball situation.
I have to measure the square footage of a building that is in the shape of an ellipse that has been truncated on either end of the major axis, so that the ...
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Understanding the Formulas for Cartesian to Hyperspherical Coordinate Transformation
I am in the process of coding a function to convert Cartesian coordinates to hyperspherical coordinates. However, I've encountered some confusion regarding the transformation formulas.
On the ...
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Regular Pentagon, Geometry
Can someone help me to find the value of $x$? I wish I could share my attempted solution but I really couldn't develop something interesting to share. I know that the value of the internal angle are $...
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How many points can be placed in a many dimensional space at a specific cosine distance away from each other?
Imagine I have a n dimensional vector space, and I pick a random unit length vector inside of this space, I now want to continue adding unit length / norm vectors ...
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Necessary condition for the product of two rotations of the plane to be a rotation
I know that any rotation of $\mathbb{R}^2$ can be expressed as the product of two reflections in non-parallel lines, and hence the product of two rotations can be written as $R_{L_4}R_{L_3}R_{L_2}R_{...
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Computing the distance from a point to an intersection of two hyperplanes
Consider the hyperplanes $\{x | P_1^T \cdot x + q_1 = 0\}$ and $\{x | P_2^T\cdot x + q_2 = 0\}$ in $\mathbb{R}^n$. Let $C \in \mathbb{R}^n$ and we want to compute te distance from $C$ to the ...
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geometry symmedian problem EGMO 4.29 [closed]
Let ABC be a triangle. The tangents of (ABC) from B and C meet at X. AX meet (ABC) at K. Show that BC is the B-symmedian of $\bigtriangleup ABK$ and the C-symmedian of $ \bigtriangleup ACK.$
This ...
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Parametric Equation of an $(n-2)$-Sphere in n-Dimensional Space on the Hyperplane $x_1 + x_2 + ... + x_n = 0$
Given a circle in a 3D space centered on the plane $x+y+z=0$, its parametric equation can be represented as:
$$
\left( \sqrt{\frac{1}{2}}\cos(t) + \sqrt{\frac{1}{6}}\sin(t), -\sqrt{\frac{1}{2}}\cos(t) ...
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Is euclidian distance an axiomatic concept or is it provable?
So euclidian distance is defined (in space $R^m$) for points $A(a_1, 1_2,...,a_m)$ and $B(b_1, b_2,...,b_m)$ as $d(A,B)=\sqrt{(a_1-b_1)^2+(a_2-b_2)^2+...+(a_m-b_m)^2}$.
It also can be thought of as ...
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Let $ABCD$ be a convex quadrilateral with $AD = BC$ and $∠A + ∠B = 120°$ . Prove triangle formed by midpoints of $CD$, $AC$ and $BD$ is equilateral.
Let $ABCD$ be a convex quadrilateral with $AD = BC$ and $∠A + ∠B = 120°$. Let $E$ be the midpoint of the side $CD$ and let $F$ and $G$ be the midpoints of the diagonals $AC$ and $BD$, respectively.
...
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Regular polygons: how to get random point position from two consecutive perpendicular projections.
Given a regular polygon with $n$ sides and a random point $P$ inside the polygon, we will draw $n$ perpendicular projections from this point to the $n$ polygon sides. Denote the point of intersection ...
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Inscribed angle theorem for hyperbolic sectors
I did the following easy exercise in 2020:
Points $A$, $B$, and $C$ are on a circle with centre $O$.
The lines through $O$ parallel to $AC$ and $BC$ intersect the circle at $D$ and $E$.
Then the area ...
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Show that $M$ is the midpoint of $JI$
*$H$ is the orthocenter of $\triangle ABC$
I don't even know where to start, it appears that point $E$ has nothing too special about it, line $LI$ is too mysterious to me, I couldn't turn it into a ...
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