For questions about geometric shapes, congruences, similarities, transformations, as well as the properties of classes of figures, points, lines, angles.

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0
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1answer
18 views

How do I get the start and endpoint of a line using the middle point and the angle?

I have a line that goes from P1 to P2 in a 2D space. I have the location of the middle point of that line, and the angle of inclination of the line. The thing is that I don't know the length of the ...
-1
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2answers
66 views

Find the Locus of the Orthocenter

Vertices of a variable triangle are $$(3,4)\\ (5\cos\theta,5\sin\theta) \\ (5\sin\theta,-5\cos\theta) $$ where $\theta \in \mathbb R$. Given that the orthocenter of this triangle traces a ...
1
vote
3answers
56 views

Calculate tangent point on ellipse

I'm trying to find a tangent point on an ellipse. Trying a lot, using answers found a.o. on this site, but obviously doing something wrong as I'm not getting any good results. I've added a sketch, to ...
0
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2answers
36 views

Prove the median and the altitude drawn to the hypotenuse make an angle congruent to the difference of the acute angles of a right triangle.

How would I go about proving this: In a right triangle, the median and the altitude drawn to the hypotenuse make an angle congruent to the difference of the acute angles of the triangle. One ...
0
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1answer
33 views

Find length of side

I tried to solve this problem ... but i can't find answer. Anyone can help me? EBC=90 & DCB=90 & AHC=AHB=90
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0answers
39 views

Overlap of Planets in Elliptical Orbit

I'm investigating further into my orbital overlap problem. I've already looked into the overlap ($0°$ angle between the two orbits) of two planets in a circular orbit around the sun. I'm now trying ...
0
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1answer
29 views

Angular Velocity around an ellipse

I'm investigating into the angular velocity of a planet in its elliptical orbit. I have these variables defined: speed of planet. speed of planet at perigee and apogee. length of orbit. ...
0
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1answer
26 views

Geometry: how to get the radius of a circle with just a chord of the circle

Answer: Able to find area with statement 1) The length of the segment AB is 10 Above are two statements asking whether I can solve for the shaded areas given the information they have provided. ...
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0answers
13 views

Sylvester–Gallai theorem in Space

Does this statement true of false? Let $P$ be a set of finite points in space,not all of them are in a plane,and any three points are not a line.Can we always find a plane just pass through three ...
3
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2answers
30 views

Find sectors in a racing line.

Consider the following photo: To magnify image, right click and select open-image in new tab or something similar The photo above is a random race circuit data I've collected. What I'm trying to ...
0
votes
1answer
13 views

Positioning a face of a regular tetrahedron normal to a remote point

I have a problem in which I have a regular tetrahedron that can rotate about its centre with all degrees of freedom. I then have a point, generated at random, to which I wish to align the closest ...
3
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0answers
49 views

Problem about cyclic quadrilaterals

In cyclic quadrilateral ABCD, let E, F, G, H be the orthocenters of triangles BCD, CDA, DAB, ABC, respectively. Prove that EFGH is cyclic. Progress So far, found that if E is orthocenter of BCD ...
2
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1answer
63 views

How does the Möbius group act on circlines?

This is a continuation of my earlier, rather vague question. I am interested in studying the action of the Möbius group $PGL(2,\mathbb{C})$, on the circlines in the extended complex plane $\mathbb{C} ...
0
votes
1answer
10 views

prove that $|x^2(y − z) + y^2(z − x) + z^2(x − y)| < xyz.$

If x, y, z are the sides of a triangle, then prove that $|x^2(y − z) + y^2(z − x) + z^2(x − y)| < xyz.$
1
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1answer
27 views

Is an open disk complete?

See the definitionS of complete surface, first definition: without edges. second definition: Any line segment can be continued indefinitely. By the first, open disk seems to be a complete surface, but ...
5
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0answers
146 views

From the viewpoint of modern geometry, is there a “best” definition of the term “triangle”?

I can think of at least six different possible definitions of the term "triangle" in Euclidean geometry. a subset of $\mathbb{R}^n$ that can be expressed as the convex hull of three or fewer points. ...
3
votes
1answer
56 views

How do you work with the space of circles on the sphere considered as the projective line?

I'm trying to prove some things about the action of the Möbius group on the "circlines" in the extended complex plane, ie. circles on $\mathbb{C}P^1$. I find that while I have a good grip on Möbius ...
0
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0answers
17 views

Symmedian and bisectors meet at the diagonal.

Let $A$, $B$, $C$, and $D$ be points on the same circle, and let the bisectors of the angles $\angle ABC$ and $\angle ADC$ intersect on the diagonal $AC$ at point $K$. Let $BD$ intersect $AC$ in $P$ ...
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0answers
19 views

Oval surface with $ K \ge 1 $ which is unit sphere

Let $ S$ oval surface with $ K \ge 1$ . If there is exist an open unit sphere interior of $ S$, then $S$ is unit sphere...Can anyone give me an idea of the solution...thanks in advance...
0
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1answer
16 views

Can you further explain or clarify the proof of this?

I am trying to prove that $AG = 2(GD)$, given $AD, BE$ and $CF$ are 3 medians meeting at point $G$ of a triangle $ABC$. I found this website that seems to show what I want to prove: ...
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0answers
20 views

Geometric question about the median and another line.

Let AM be the median of the triangle ABC. Let N be also on BC, but its not on the same spot as M, so that the angles BAM and CAN are equal. What is the line AN called and what properties does it have? ...
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2answers
22 views

Find a point C on an infinite line AB which, when connecting two other points M and N, would form congruent angles

On an infinite line $AB$, find a point $C$ such that the rays $CM$ and $CN$ connecting $C$ with two given points $M$ and $N$ situated on the same side of $AB$ would form congruent angles with the ...
0
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2answers
27 views

fixed points of an affine transformation is unique iff $1 \notin SP(\vec{f} )$

Let $f$ be Affine transformation from $E$ to $E$ (always we assume it finite dimensional ) and $\overrightarrow{f}$ is the linear mapping associated to $f$. Then the map $f$ has a unique fixed ...
0
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2answers
38 views

analytical proof for Jacobi identity

I need proof I am in need of major assistance with a homework problem I have been working on $$\vec a(\vec b\times \vec c)+ \vec b(\vec c\times \vec a)+ \vec c(\vec a\times \vec b)=0$$
1
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1answer
28 views

How to show that the cross-ratio is invariant under Moebius transformations without using generator matrices of $\text{GL}(2, \mathbb{C})$

I am given the following problem set: Show that the cross-ratio is invariant under Moebius transformation, meaning that $$D \left(L_g(z_1),L_g(z_1),L_g(z_1),L_g(z_1)\right) = D(z_1,z_2,z_3,z_4) ...
2
votes
1answer
27 views

Slice an ellipsoid into equally thick slices for maximal surface

After seeing a colleague slicing a nearly ellipsoid piece of ginger for his cup of tea into almost equally thick slices to get more surface area (so the tea would suck out the ginger taste better), i ...
-2
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0answers
56 views

Bisectors of opposite angles of a circular quadrilateral meet at the diagonal.

Let ABCD be a circular quadrilateral so that the bisectors of angles ABC and ADC meet at the diagonal AC. Let M be the midpoint of AC. Let q be a line parallel with the side BC so that q passes ...
0
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1answer
48 views

“Rigid” Riemannian metrics

What do we mean when we say that a Riemannian metric $g$ is rigid? For example, the Eguchi-Hanson metric is rigid as an Einstein metric. Any help is appreciated!
1
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1answer
15 views

Disjoint sets of vertices of a polygon

This is the question: Suppose that the vertices of a regular polygon of 20 sides are coloured with three colours – red, blue and green – such that there are exactly three red vertices. Prove that ...
3
votes
2answers
28 views

Proof related to circumcircle of triangle

I have a triangle $ABC$ with incenter $I$. $AI$ extended meets the circumcircle of $ABC$ at $M$. Prove that $CM=BM=IM$. I was able to prove that $CM=BM$ taking advantage of the fact that the ...
1
vote
1answer
60 views

What is the etale fundamental group of Z((x))?

I know the fundamental group of $\mathbb{Z}$ is trivial, and the prime candidate for finite etale covers involving the $x$'s is the endomorphism sending $x\mapsto x^n$, but that's ramified at all ...
5
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0answers
58 views

How big is a tetrahedron?

Let $T$ be a tetrahedron with volume $vol(T)$ and edge lengths $a,b,c,d,e,f$ and let $sum(T) = a^3 + b^3 + ... + f^3$. We wish to compare $vol(T)$ with $sum(T)$. [ IMO (1961 #2 ) handles the case of ...
0
votes
1answer
13 views

Is the following convex geometry relating intersection and set averages true?

Let $X$ and $Y$ be two convex cones, and denote by $(1/2)*X+(1/2)*Y$ the Minkowski average of $X$ and $Y$ (i.e., $\{z:z=(1/2)*x+(1/2)*y,x\in X,y\in Y\}$). Then $$X\cap Y \subseteq(1/2)*X+(1/2)*Y$$ Is ...
0
votes
1answer
46 views

Proving geometry using complex numbers?

Consider the following figure: Let $A(z_1),B(z_2),C(z_3),E\equiv P(z),O(\mathtt{0})$Q(-z), I need to prove BQ=AC, I can prove it anyways but using complex numbers. Anyways what I tried is as ...
0
votes
1answer
14 views

The point on a conic furthest from a given line

To solve the problem of finding the point on an ellipse furthest from a given line: Finding the point on an ellipse most distant from a given line, I was given the suggestion to use the fact that the ...
1
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0answers
31 views

Finding the point on an ellipse most distant from a given line

$\mathrm C$onsider an ellipse with the origin as its centre, i.e., of the type $$\frac {x^2} {a^2} + \frac {y^2} {b^2} = 1$$ and a line joining two points on the ellipse. $\mathrm T$he problem is to ...
1
vote
2answers
34 views

A conjecture about lines and points in the plane

Let $ l_1, l_2, \ldots$ be an infinite sequence of lines in the plane, and let $(a_1, b_1), (a_2, b_2), \ldots$ be an infinite sequence of pairs of points such that $a_i, b_i \in l_i$ and $a_i \neq ...
1
vote
1answer
35 views

An interior point in the triangle

Suppose $P$ is an interior point of a triangle $ABC$ and $[AP]$, $[BP]$, $[CP]$ meet the opposite sides $[BC]$, $[CA]$, $[AB]$ in $D$, $E$, $F$ respectively. Find the set of all possible values of the ...
0
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0answers
15 views

Pasch's axiom in the hyperbolic plane

Q: I am asked to show that pasch's axiom holds true in the hyperbolic plane. Pasch's axiom Idea: I was thinking of about a circle $\Gamma$ whose centre is the origin of the complex plane. ...
1
vote
3answers
58 views

What is essence of second order derivative?

Let $u(x)\in C^2(R)$ is a real function. so: $$ u'(x)=\lim_{\Delta x\rightarrow 0} \frac{u(x+\Delta x)-u(x)}{\Delta x} $$ And: $$ u''(x)=\lim_{\Delta x\rightarrow 0} \frac{u'(x+\Delta x)-u'(x)}{\Delta ...
2
votes
2answers
22 views

Maximum ratio between diameter of shape and diameter of enclosing circle

For a 2-dimensional closed convex shape $C$, define: $d(C)$ = the diameter of $C$ (the largest distance between two points in $C$). $D(C)$ = the diameter of the smallest circle containing $C$. ...
0
votes
1answer
12 views

Dimensions and their Measurements.

The basic measurements are, well, basic: A line had Length A square has Area A cube has Volume But my question is simply, what goes in the ?s in the next list? A tesseract has ? A ...
0
votes
1answer
18 views

Area defined by cylindrical coordinate equations

I'm supposed to find the area of a space defined by these equations. r=3 0$ \le $$ \theta $$ \le $$ \textstyle\frac{ \pi }{2} $ 0$ \le $z$ \le $2 I tried applying A = $ \int^b_a 0.5r^2\,d\theta $, ...
1
vote
1answer
40 views

Given radius, and many vertices on it, how can I find center of a sphere?

I have a sphere, I know its radius. I also have the coordinates of 500 vertices which are on the sphere. How can I find the center coordinates of a sphere? Is there an easy way to do that? Thanks.
2
votes
4answers
42 views

Area between two polar areas

I could use some help with this problem. Let a be a constant. Find the area that stays inside both the circle $r = a$, and the cardioid $r = a(1-\sin\theta)$. I tried to find the point of intersect ...
0
votes
2answers
51 views

To find two sides of a triangle when it is circumscribed a circle

A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6 cm respectively. Find the ...
1
vote
1answer
21 views

Problem with spherical coordiantes

So I'm trying to convert $ x^{2} $+$ y^{2} $ = 9, into spherical coordinates, here is my process : $ x^{2} $+$ y^{2} $+$ z^{2} $ = 9+$ z^{2} $ $ \rho^{2} $ = 9 + $ \rho^{2} $$ cos^{2} $($ \phi $) ...
4
votes
2answers
56 views

How many lines are needed to create 6 triangles on W?

Basically, the question started with a little argument I had with my friend. My friend said he thinks it's possible to draw only 2 lines on the letter "W" and make 6 triangles, and I played around ...
0
votes
1answer
29 views

Converting from Non-basis coordinates to XYZ. Solving system of equations. Error volume

I have multiple points in 3D space. Each point has the distances to 3 points. Those 3 points are: (50,0,0) (0,50,0) (0,0,50) Lets call those distances $dx,dy,dz$ I want to find $x,y,z$ of those ...
0
votes
1answer
16 views

Orbital Motion overlap

What's a possible way of finding time t for an overlap of three objects going around a circular orbit around a common fixed center. So like the solar system in 2D, How can I find the time for when the ...