Questions tagged [geometry]
For questions about geometric shapes, congruences, similarities, transformations, as well as the properties of classes of figures, points, lines, and angles.
52,595 questions
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Proving hermite conditions of a polygonal patch
In parametric lines, constructing the standard hermite basis is trivial.
We have two points $p_1, p_2$ and two tangent vectors $t_1, t_2$. Thus we have 4 unknowns that will be sampled, for that we ...
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Is my Euclidean-style proof valid? It is for a summation of infinitely many line segments equaling to a finite length without calculus, or limits. [closed]
I have attempted to prove that the sum of infinitely many quanities can still equal a finite quantity without using calculus, measure theory, or any other modern mathematical tool such as set theory.
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Find the locus that divides the line joining a point and any arbitrary point on the circumference of a circle in a fixed ratio [closed]
Given a point and a circle, find the locus of points that divide the line joining the given point and an arbitrary point on the circumference of the circle in a fixed ratio.
(If A is a point and C(O, ...
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Basic Proportionality Theorem/ Thales Theorem [closed]
The Basic Proportionality Theorem seems so obvious but the construction to prove it (drooping perpendicular to equate areas) is not at all obvious to me. Can anyone tell how to prove this Theorem in a ...
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geometric problem for spiral similarity
We consider a quadrilateral $ABCD$ inscribed in a circle $\omega$. Let $P$ be a point inside $\omega$ and the following equalities are satisfied
$$\angle PAD = \angle PCB,\ \angle ADP = \angle CBP.$$ ...
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Find out the distance between centers of two intersecting semi-ellipses $x^2/a^2+y^2/b^2=1$.
There are two identical semi-ellipses, one with center at the origin $O$, $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, and the other at $R$, $\frac{(x-d)^2}{a^2}+\frac{y^2}{b^2}=1$.
Find out the distance $d$ ...
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Is this a known theorem?
Reference image ^^^
Edit: a person has answered this and I have rediscovered Euclid's Elements, Book 13, Proposition 15.
Ok so I think I might have found a new theorem or maybe rediscovered an old one....
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How would you determine a circle diameter using three smaller known circle diameters that all fit neatly within this circle. [closed]
Given three circles with diameters of .623", .687", and .719" that fit snugly within a circle of diameter D, what is D? What is the mathematical formula for this?
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Find the value of y in this geometric figure
Find the value of $y$ in the following geometric figure, as a function of $v_1$, $v_2$, $x$ and $H$. All angles that visually seem to be $90°$, are.
I was asked to share what I tried, so here it goes....
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What is the measure of the angle between the two diagonals of this trapezoid?
The attached figure represents a trapezoid ABCD with four angles indicated.
My objective is to calculate the angle x formed by the two diagonals AC and BD.
Using GeoGebra, I found that x is almost 106°...
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A subset of a rectangle that 'blocks' every curve that goes from right to left must connect upper and lower sides
I'm trying to find a proof for the following assetion: Given a rectangular region $R$ and a subset $A$ of $R$, if every curve that starts at the left side of $R$ and ends at the right side intersects $...
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How to find the maximum length of chord cut by a right angle inside a circle
How to find the maximum length of chord AB in the figure below?
P is a fixed point inside circle centered at O (P is not O). PA and PB form a right angle. Imagine this right angle rotates inside the ...
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Over $\mathbb R^2$, given a tool that can $n$-sect an angle for any $n$, for which $n$ can you construct the $n$th root of any given $x > 0$?
It's classically known that you cannot, say, construct the $n$th root of $2$ for $n \ge 3$ and $n$ not a power of $2$ with just ruler and compass. However, recall taking $n$th roots and the Chebyshev ...
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Exercise in Acoustic Doppler Effect
I am looking for some guidance on the second part of a geometry type problem which I have given working on and described the next parts below (likely with an error). I have given multiple attempts but ...
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Determine the reflection line about which two points reflected will lie on two given lines respectively
You're given two lines in the $xy$ plane, let's say
$ Line 1: a_1 x + b_1 y + c_1 = 0 $
and
$ Line 2: a_2 x + b_2 y + c_2 = 0 $
In addition you're given two points $P = (p_1, p_2) $ and $Q = (q_1, q_2)...
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Proving a Claim about four mutually tangent unit spheres
Prove the Claim about four mutually tangent unit spheres :
(1) The centers of each sphere lie at the vertices of a regular tetrahedron of edge length $2$
(2) Their points of tangency lie at the ...
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Is a mapping that maps parallelograms to parallelograms a collineation (on an affine plane)? [closed]
It is easy to show that in a $(\mathcal{P},\mathcal{L})$ affine plane any collineation maps any parallelogram to a parallelogram. But is it true that if a $\mathcal{P} \to \mathcal{P}$ bijection maps ...
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Congruent triangle to a given one, with its vertices at specified distances from the original triangle vertices
You're given $\triangle ABC$ with known vertices in the $xy$ plane. The coordinates of $A,B,C$ are known. Now, given three distances $d_1, d_2, d_3$. You want to determine all congruent triangles $\...
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Find the ratio $\frac{AC}{BC}$ given a specific configuration of equilateral triangles around a right triangle (need Euclidean geometry approach)
I encountered a geometry problem involving a right-angled triangle and several constructed equilateral triangles. I am trying to solve the second part of the problem (Case 2 in the image).
Continues ...
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Prove that the center of the sphere, the centers of two small circles, and their single common point lie in the same plane
Two circles are drawn on a sphere, having a single common point.
Prove that the center of the sphere, the centers of both circles, and their common point lie in the same plane.
This is equivalent to :...
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Usage Of $T, S $ and $ S_1$ [closed]
In conic sections there are many formulas that involve the three forms:
$S$, $T$, and $S_1$, or combinations of these. But I have some questions regarding them.
For a general conic:
$S \equiv ax^2 + ...
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What is the most concise complete definition of a rigid framework?
From what I've seen, the key characteristic of a rigid framework in a polygon is that the sides of the polygon, once set, force the distance between every pair of vertices to remain constant. Is "...
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Relationship between distance from chords and distance to tangent
Chord $AB$ is drawn in a circle. A tangent is drawn to the circle through point $C$. Let the distances from points $A$ and $B$ to this tangent be $a$ and $b$. Find the distance from point $C$ to chord ...
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Find the ratio of side lengths of two equilateral triangles given a midpoint condition
Problem Statement:
As shown in the diagram below, we have two equilateral triangles, $\triangle ABC$ and $\triangle ADE$, sharing a common vertex $A$.
We construct a line connecting vertices $B$ and $...
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Vieneuous Triangle Theorem: Point on median with half-altitude distance sum in isosceles triangle [closed]
Proposing the Vieneuous Triangle Theorem (Nov 24, 2025, from Vietnam).
Theorem: In isosceles $\triangle ABC$ ($AB=AC$), median $AD$ is from $A$ to base midpoint $D$. There exists unique $P$ on $AD$ s....
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Golden ratio from Isosceles
I have the following Isososcles Triangle, and after some headscratching i managed to figure out the golden ratio. By Comparing the ratio of the longer segment to the shorter segment relative to the ...
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How far can an infinite number of unit length planks bridge a right-angled gap?
Problem: You are standing at the origin of an infinite flat earth. One quadrant of the earth is gone, leaving an infinite abyss. You have an infinite supply of unit-length zero-width planks. How far ...
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What is the length of the height AH?
I'm trying to find a problem about right triangles with a minimalist statement that isn't too obvious. Here's what I've come up with :
ABC is an A–right triangle, H is the orthogonal projection of A ...
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Fermat-Torricelli Weighted Point
I am in the process of designing a global trajectory program for civil aircraft. Two aircraft depart from their airports, join together to create a formation, then later separate and land at their ...
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Is every simple curve the preimage of a regular value?
A set $C\subset \mathbb{R}^2$ is a simple curve if $\forall p \in C$ there exist $I$ an open interval, $V$ an open set in $\mathbb{R}^2$ containing $p$; and $\alpha \colon I \rightarrow \mathbb{R}^2$ ...
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A question about rotating and shifting a triangle
Suppose you're given a triangle $\triangle ABC$ in the $xy$ plane, with known side lengths. Now you rotate and shift this triangle to some other position/orientation generating a congruent triangle $\...
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Similarity argument in right triangle with perpendiculars to sides
I have a right triangle $OAB$ with right angle at $O$, and let
\begin{equation}
OA = L, \quad OB = 1.
\end{equation}
Let $a$ be a point on $OA$ and $b$ a point on $OB$. From these points, ...
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Find $\angle C$ given the relation $a^4 + b^4 + c^4 = 2a^2c^2+2b^2c^2$ [closed]
In a $\triangle ABC$ with sides $a, b, c$, the following relationship holds:
$$a^4 + b^4 + c^4 = 2a^2c^2+2b^2c^2$$
I need to determine the possible values for angle $C$.
My Attempt:
I suspect this ...
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Help: Proof of Ptolemy's Inequality with Complex Numbers
This was the original problem statement:
Let $ABCD$ be a quadrilateral, where $A, B,C$ and $D$ are points in anti-clockwise direction corresponding to $z_1, z_2, z_3, z_4\in\mathbb{C}$ respectively. ...
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Find the length of the side DQ
In rectangle $ABCD$, points $P$ and $Q$ lie on sides $AD$ and $DC$ respectively, such that $AP = 2 \times DQ$.
Given that $AB = 5\,\text{cm}$, $BC = 10\,\text{cm}$, and the area of quadrilateral $BPQC$...
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triangle construction by using Eyclid Geometry [closed]
costruct a triangle ABC if BC=8 cm, AB+AC=15 cm, and Ha=3 cm,where Ha is the height from A to BC, using Euclid Geometry only.
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Perpendiculars passing through diagonal intersection in a quadrilateral formed within a square
Let $ABCD$ be a square with points $F \in BC$ and $H \in CD$ such that $BF = 2FC$ and $DH = 2HC$.
Construct:
Line through $F$ parallel to $AB$, meeting $AD$ at $E$
Line through $H$ parallel to $BC$, ...
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Solving quintic equations with PowerPoint shapes
We are attempting to measure roots of quintic equations in Powerpoint. This is a mathematical curiosity inspired by Dr. Zye's recent video on making flags in Powerpoint. If you are interested in the ...
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Show that it is impossible to construct a segment of length $\sqrt[3]{2}$ (the cube root of 2) with a straightedge and compass from a unit segment? [closed]
I came across this statement, that it is impossible to construct a segment of length $\sqrt[3]{2}$ (the cube root of 2) with a straightedge and compass from a unit segment.
I know that one can ...
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Can an isosceles triangle with a $60^\circ$ angle be proven equilateral independently of the triangle angle sum theorem?
There is a famous theorem in elementary geometry:
Theorem. An isosceles triangle with a $60^\circ$ angle is equilateral.
Two cases of this theorem are depicted below. I consider any (or both) of ...
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High School Geometry Problem involving Golden Ratio
This is picture of the following problem. Here are the steps I took. Firstly, I was confused by the similar triangle statement. By doing some angle chasing I believe that ∆ABD~∆ACB. So for the rest of ...
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Angle between tangents of a hyperbola
Let a hyperbola with semi major axis length $a$ and shortest radius $r_p$ be given. For $r\geq r_p$ find angle $\gamma$ between the tangent at distance $r_p$ and the tangent at distance $r$ from the ...
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So we know about the Cartesian coordinate system, but how to naturally arrive at it when going on a quest to identify a location? [closed]
You would need to use the concept of distance and direction in order to derive the coordinate system with the three perpendicular axes from the first principles.
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On the convolution identity of a sub arc of circle and the open set which is thickened epsilon amount of another subarc in circle.
Let $\sigma_I=\{e^{2i\pi t}:t\in I\}$ and $\sigma_J=\{e^{2\pi i t}:t\in J\}$ be two disjoint subarcs on the the first quadrant of unit circle of arc length $\theta$, where $I,J\subseteq [0,1/4]$ of ...
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Counting disjoint $k$-tuples of lines in $\mathbb F_q^n$
Let $k$ be a positive integer. How many $k$-tuples of disjoint lines are there in $\mathbb F_q^n$? Here two lines are disjoint if they do not share a point in $\mathbb F_q^n$.
I was wondering because ...
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Can you find a great circle with only a compass?
Thinking about the construction of temari balls got me thinking about how one might begin the marking portion of the process, particularly how to build a great circle in the first place. To make the ...
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Cutting a Möbius strip in thirds. Why are the resulting strips interlinked?
It is well known that cutting a Möbius strip "in half" down the middle results in a band with two twists, homeomorphic to a cylinder. See this question for example.
If instead, one begins ...
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Finding the angles of a non-equilateral $\triangle ABC$ with centroid $G$ such that $\angle GAB=\angle GCA=30^\circ$
In the attached figures, $G$ is the centroid of $\triangle ABC$.
When this triangle is equilateral ( fig 1) , it is obvious that each of the two angles shown in this figure measures $30^\circ$.
Using ...
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The only area–preserving linear symmetry of a strictly convex body fixing a boundary point is the identity?
If $K\subset\mathbb R^2$ is strictly convex, $T\in SL(2,\mathbb R)$ is linear with $T(K)=K$, and $T$ fixes some boundary point $p\in\partial K$, must $T$ be the identity?
Without strict convexity, we ...
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Can a hemispherical surface be flattened if tearing is allowed but no stretching is allowed in the interiors
Can a hemispherical surface be flattened if tearing is allowed but no stretching is allowed in the interiors ?
A flattening, in this case, would be a continuous mapping from the hemispherical surface ...
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