Questions tagged [geometry]
For questions about geometric shapes, congruences, similarities, transformations, as well as the properties of classes of figures, points, lines, and angles.
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Looking for an example of a real (hyper)cubic surface with special restriction
Need help looking for an example of a cubic surface over $\mathbf{R}$: it has two connected components, one of the components is convex. Here's my thoughts so far:
The model in my head is the ...
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Help me with this problem please [closed]
Let ABC be a triangle with AB = 18, BC = 24, and CA = 20. D is placed on AB such that AD = 15. E is placed on BC such that EC = 20. Call the intersection of lines AE and DC point F. Compute:
$(\text{...
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Orthocenter: The "Bad Boy" of Distinguished Points in a Triangle
It is a well-known fact that the altitudes of a triangle $ABC$ (with vertices $A,B,C)$ intersect at exactly one point, the orthocenter. The proof known to me (see eg here) involves the construction of ...
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Meaning of the German term "Schließungssatz"
Certain theorems in geometry are called Schließungssatz in German, like the theorem of Desargues and the theorem of Pappos, see for example this German wikipedia page.
I wonder what the (historical?) ...
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Equality of seemingly unique geometric arragements [closed]
I'm not a mathematician or anything so please forgive poor formatting/terminology.
Let's first agree that a cube in contact with and directly on top of an infinitely large plane has 3 states:
1: The ...
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How to calculate the probability of a random great circle in a sphere pass through a certain spherical cap?
at some point, I got lost in some calculations. So, there is a sphere and a spherical cap in it. Imagine a great circle (one that passes through the center of a sphere) is chosen at random. I want to ...
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concyclicity related to the Humpty point
$AD$ and $CE$ are the altitudes of triangle $ABC$, $M$ is the midpoint of $AC$. Circumcircles $\omega_1$ and $\omega_2$ of triangles $AEM$ and $CDM$, respectively, intersect at point $P$. $CP$ and $AP$...
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Find the ratio of the area of the triangle $PRQ$ ¸and the area of the triangle $ETA$
the problem
On the side $BC$ of the triangle $ABC$, the points $D$ and $E$ are considered such that $BD = DE = EC$. Let $M$
the middle of the segment $AD$, $BM ∩ AE = {P}, CM ∩ AE = {Q}$. $RM$ ¸and $...
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Rational quantities associated with a bicentric heptagon
For odd reasons of my own, I would like to create a math puzzle involving a bicentric heptagon (i.e., a 7-sided polygon which has both an inscribed and a circumscribed circle). The puzzle is to be ...
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How to express an angle between two angle bisectors in interior angles of a convex quadrilateral?
Given a convex quadrilateral $ABCD$, I would like to express the angle between angle bisector of internal and external angles of the opposite vertices in interior angles of $ABCD$.
Here is the drawing ...
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Find the segment "DC" in the obtuse triangle below
In an obtuse triangle $ABC$, obtuse at $B$, the internal bisector $AD$ is drawn and in $AC$ the point $"q"$ is taken such that $m∡𝐴𝐷𝑄=90^𝑜$.
Calculate $DC$. If: $AQ = 10$ and $AB = BC$.
$Answer:...
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How to solve this optimization problem explicitly with Lagrange multipliers? [closed]
In reference to this other question of mine: The maximum area of a pentagon inside a circle and under the advice of one of the people who answered (@intelligent pauca), I decided to ask this new ...
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Pentagon $(ABCDE)$ is inscribed in a circle of radius $1$. If $\angle DEA=\angle EAB = \angle ABC$ and $m\angle CAD=60^{\circ}$ and $BC=2AB$.
Pentagon $(ABCDE)$ is inscribed in a circle of radius $1$. If $\angle DEA=\angle EAB = \angle ABC$ and $m\angle CAD=60^{\circ}$ and $BC=2AB$.
Compute the area of $(ABCDE)$
Firstly I noticed that ...
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Find the minimum value of the sum of the squares of the distances from M to the lines AB, AC and BC
The problem
Let ABCA'B'C' be a regular triangular prism with base edge $AB = 2 \sqrt{3}$
cm and height
AA' = 1 cm. If M is a point in the plane of the triangle A'B'C'
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, then the minimum value of the ...
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Is this proof of the angle bisector theorem known?
Given a triangle $ABC$, let $D$ be the point of intersection of the side $BC$ with the bisector of the angle $A$. Then $|AB|/|AC|=|DB|/|DC|$.
This statement is known as the angle bisector theorem.
Is ...
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