Questions tagged [geometry]
For questions about geometric shapes, congruences, similarities, transformations, as well as the properties of classes of figures, points, lines, and angles.
34 questions from the last 7 days
41
votes
7
answers
2k
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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 ...
7
votes
1
answer
925
views
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....
- 123
8
votes
1
answer
460
views
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 $...
- 333
5
votes
5
answers
268
views
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 ...
3
votes
4
answers
243
views
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 ...
- 133
3
votes
3
answers
195
views
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 ...
- 6,461
1
vote
3
answers
191
views
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°...
- 1,806
2
votes
4
answers
229
views
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 ...
- 1,806
5
votes
1
answer
170
views
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 $\...
- 188
4
votes
1
answer
177
views
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....
- 41
4
votes
2
answers
96
views
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 :...
- 6,461
4
votes
1
answer
113
views
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 $...
4
votes
1
answer
75
views
Find a synthetic proof to an old problem .
I found this problem in a French paper translated from Arabic in 1927 by an author named Al Bayrouni . I wonder if it can be found in one of Archimedes' works . Here is the statement :
ABC is a ...
- 1,806
2
votes
1
answer
64
views
Find out the distance between centers of two intersecting semi-ellipses $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$.
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 $d$ ...
- 1,290
1
vote
2
answers
71
views
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)...
- 188
0
votes
0
answers
94
views
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 ...
- 79
0
votes
1
answer
81
views
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 ...
- 3
2
votes
1
answer
87
views
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 $\...
- 188
1
vote
1
answer
45
views
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, ...
- 609
2
votes
1
answer
43
views
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 ...
3
votes
0
answers
63
views
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 ...
- 111
0
votes
0
answers
60
views
+50
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 "...
- 263
1
vote
0
answers
72
views
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 + ...
- 119
2
votes
1
answer
114
views
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 $\alpha \colon I \rightarrow \mathbb{R}^2$ ...
0
votes
0
answers
50
views
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 ...
0
votes
0
answers
46
views
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?
1
vote
0
answers
73
views
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 ...
- 6,461
0
votes
0
answers
51
views
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....
0
votes
0
answers
37
views
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, ...
1
vote
0
answers
47
views
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
$$\angle PAD = \angle PCB,\ \angle ADP = \angle CBP.$$ ...
- 5,246
0
votes
1
answer
46
views
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 ...
0
votes
0
answers
30
views
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.
...
0
votes
0
answers
35
views
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 ...
- 81
0
votes
0
answers
4
views
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 ...
- 3,857