Questions tagged [geometry]
For questions about geometric shapes, congruences, similarities, transformations, as well as the properties of classes of figures, points, lines, and angles.
52,639 questions
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Equilateral triangle from midpoints and circle intersections
Problem:
Given equilateral triangle $ABC$ inscribed in circle $(O,R)$, let $K,L,M$ be the midpoints of $AB,BC,CA$ respectively. If line $KM$ intersects the circle at point $D$ and line $KL$ intersects ...
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Tangential quadrilaterals with the property: The incenter equals the lamina centroid (note: not the vertices centroid!)
For symmetry reasons it is clear that rhombi do have this property. It is also not difficult to find some special kites having this property. And here the open problem begins: I am rather sure that no ...
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Simplex with all altitudes equal
For an $n$-simplex in $\mathbb R^n$, if all of its altitudes are equal (i.e the distance from any vertex to the opposide facet is equal), must it be vertex-transitive (i.e. the stabilizer of the ...
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Help for proof/counter-proof regarding convex hulls in $\mathbb{R}^2$
I am a pre-university student, and I recently stumbled upon an Advent of Code programming problem (2025 Day 9), whereby one must find the biggest axis-aligned bounding box (AABB), by area, amongst all ...
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How to find the x coordinate of the center of a circle given y coordinate, radius, and a coordinates of a random point on the circle? [closed]
I'm struggling an embarrassing amount with this. In my case:
Center is at $(C_x,20.82)$
Radius is 0.95
The coordinate of a random point on the circle is $(0.38,20.385)$
It feels like there should only ...
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Maximum number of points in an open unit square with pairwise distances greater or equal than $1/2$
I was reading this question about whether 4 points in a unit square can have mutual distances $> 1$.
This made me wonder about the case where the distance threshold is halved.
Let $S$ be the open ...
- 1,067
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A single-sided infinite surface, does that exist? [closed]
I just read a question about the surface of a sphere, and it just hit me:
The surface of a sphere is infinite: in every direction you choose, you can go on forever. On the other hand, the surface of ...
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(Ahlfors) Prove analytically that the midpoint of parallel chords to a circle lie on a diameter perpendicular to the chords
I was going through Ahlfors' Complex analysis when I came across the question in the title. This question is quite easy to answer geometrically (just show that the diameter perpendicular to the two ...
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Determine the surface area of Archimedean solids with radius 1
For Archimedean solids with given edge length (say, $1$), it is easy to determine the surface area of the solid. One can also compute tha radius (with which I mean the radius of the circumsphere) of ...
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Prove $BE^2 = AE \cdot AB$ in an isosceles right triangle with a perpendicular condition
I encountered this geometry problem involving an isosceles right-angled triangle and I am looking for a synthetic geometric proof (Euclidean geometry), as my attempt using coordinates became quite ...
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Minimizing Side BC with 2 Perpendicular and One Equal Side Length Constraints(Find track of C)
Question
As shown in the figure, in $\triangle ABC$, the length of side $AB$ is $2$. $BD$ is perpendicular to $AC$ ($BD \perp AC$). Point $E$ lies on the line $BD$ such that $BE = AC$ and $\angle CEA =...
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Simplex tiling used in Simplex Noise
Simplex noise involves a tiling of $\mathbb R^n$ with $n$-simplicies congruent to the simplex with vertices
$$
\left\{
M \begin{pmatrix}0 \\ 0 \\ \vdots \\ 0 \end{pmatrix},
M \begin{pmatrix}1 \\ 0 \\ \...
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Find the singular points, their multiplicities, and the principal tangents of the following complex projective plane cubics [closed]
Find the singular points, their multiplicities, and the principal tangents of the following complex projective plane cubics. (1) C : (y^2)z − x^3 − (x^2)z = 0,
(2) D:(y^2)z−(x^3) =0.
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Is there an elegant proof to this old problem of parallelism that dates back to the 19th century?
Here is an old problem from the 19th century for which I have lost all other references:
Two circles (W) and (W') intersect at two points P and Q. The two tangents to (W) from P and Q cut (W') again ...
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Does a self intersecting curve contain a simple curve
I got to this problem as part of a different problem I'm working on, not from any homework set. Any solution or reference from any field might help.
Say I have a smooth closed curve $\gamma:[0,1]\to\...
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