# 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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+50

### 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 ...
1 vote
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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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+100

### 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
1 vote
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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 ...
116 views

### 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 ...
1 vote
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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 ...
34 views

### 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 ...
44 views

### 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 ...
68 views

### 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 ...
1 vote
60 views

### 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 ...
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 ...