Questions tagged [geometry]
For questions about geometric shapes, congruences, similarities, transformations, as well as the properties of classes of figures, points, lines, and angles.
52,934 questions
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The Mathematics behind neural network [closed]
So, what's really going on with a neural network. Let's say, we are given a function real valued f of two variables $(x_1, x_2)$. Using a single layer with one neuron is just the linear approximation. ...
- reputation score 11
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Eight points in a unit cube, separated from each other by more than or equal to one unit
I try to solve that problem from math folklore (I don't know its solution by hand.)
Let eight points in a unit cube separated from each other by more than or equal to one unit be given. Prove that ...
- reputation score 6115
Score of 4
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Prove the radii of two circles are equal
I've spent a lot of time trying to prove this problem using the Sine Law, writing ratios via Thales' theorem, and looking for a homothety argument, but I still can't prove the required equality.
The ...
- reputation score 335
Score of 4
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Geometric intuition of Integration by Parts: How to visualize non-increasing functions?
I am a high school student studying calculus. While learning Integration by Parts, I tried to derive the formula geometrically rather than relying solely on algebraic manipulation.
I set up a new ...
- reputation score 61
Score of 9
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Proving Napoleon's Theorem with classical geometry
Given any triangle $ABC$. Construct equilateral triangles externally on each side of $ABC$. Let the centers of these three equilateral triangles be $X$, $Y$, and $Z$.
Prove that $XYZ$ is also an ...
- reputation score 99
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Visual proof for $4\sin(x) \sin(60^\circ-x) \sin(60^\circ+x) = \sin3x $ [closed]
I've been noodling around looking trying to find a visual proof or construction to show $4\sin(x)\sin(60^\circ-x)\sin(60^\circ+x) = \sin(3x) $ or the equivalent cosine statement. Does anyone know of ...
- reputation score 145
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How to prove this problem? [closed]
Given chord BC of a circle O, and point A inside the circle such that angle BAC<90, let ACP and ABR be an isosceles right triangle outside ABC, draw BA,CA intercepts O at E,F, let EP,FR intercept O ...
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Adding drawing to my post [migrated]
I need to add a drawing to my geometry post; how do I do it? I did not notice any button for adding figures/drawings, thus I did nothing
- reputation score 162
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1 answer
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Automorphisms of $\mathrm{PG}_2(\mathbb{F}_4)$, identity on line, and fixing $2$ points outside it
Let $\mathrm{PG}_2(\mathbb{F}_4)$ be the projective plane over the finite field $\mathbb{F}_4$.
It has $21$ points, $21$ lines, two lines intersect at unique point. Further, each line contains $5$ ...
- reputation score 4053
Score of 9
1 answer
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Concerning Ron Gordon's solution for Cleo's integral which implies the Kepler Triangle identity
As is well known, Cleo's integral has the result of
$$I=\int_{-1}^1\frac{1}{x}\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2x^2+2x+1}{2x^2-2x+1}\right)dx=4\pi\cot^{-1}\left(\sqrt{\phi}\right)$$
being the ...
- reputation score 668
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Construct a function $f(x)$ defined on $\mathbb{R}$ whose graph is invariant under a rotation of $π/3$ about the origin.
Construct a function $f(x)$ defined on $\mathbb{R}$ whose graph remains invariant under a rotation of $π/3$ about the origin.
I searched MSE for discussions on this topic, but only found results ...
- reputation score 87
Score of 4
1 answer
328 views
Planarity of spherical graphs
I am interested in the even distribution of points on a sphere and accidentally discovered next fact.
Consider the following configurations:
Vertices of platonic solids
Solutions of Thomson problem
...
- reputation score 1031
Score of 3
1 answer
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Identify triangle center
On the 50th Polish Math Olympiad, there was this problem:
Points $D$, $E$, $F$ lie on the sides $BC$, $CA$, $AB$ of triangle $ABC$ correspondingly. Incircles of $\triangle{AEF}$, $\triangle{BFD}$, $\...
- reputation score 133
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Synthetic proof of a spiral similarity problem involving tangents and reflections
Problem: Let $KJL$ be a triangle with circumcircle $\omega$. $MJ$ and $ML$ are tangents to $\omega$. $J'$ be the reflection of $J$ over $KL$. $N$ be the circumcenter of $\triangle JML$. $V$ be the ...
- reputation score 886
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Counting 3D Regions Formed by Nested Polyhedra Constructed from Face Centroids
In a previous question Counting Regions Formed by Nested Polygons Constructed from Side Midpoints, I investigated the number of 2D regions created by iteratively nesting smaller polygons inside a ...
- reputation score 377