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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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2019 circles are located on a sphere. These circles divide the surface of the sphere in how many areas?

2019 circles are located on a sphere. These circles divide the surface of the sphere into how many areas? We are not provided how the circles are engraved.
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What are some tips for framing a graph in the most readable, relevant, and aesthetic way, for secondary education mathematics?

When I say "framing," I mean things like choosing zoom, x-axis/y-axis step, horizontal/vertical shift from the origin, choosing how/when to number steps, labeling axes, as well as, purely aesthetic ...
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1answer
20 views

Help me find the coordinates of a point $9$ on a circle.

This seems easy. But it isn't. The diameter is given as $16$ and it asks you to find the coordinates of point $9$. It's tempting to say that it is $(4, 4\sqrt{3})$, but that isn't the answer. What the ...
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15 views

Can y' be introduced into a basic optimization problem?

Let's spell out the basic procedure from Calculus 1 for finding critical points. Given an equation $f(x,\ y) = g(x,\ y)$, $1)$ Take the derivative of both sides $2)$ Plug $0$ in for each ...
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1answer
13 views

Length across cuboid at an angle

I would like to calculate the change in length across a cuboid when looking at it from different angles (e.g. 10 and 20°). In the illustration below the red arrows show the length I would like to ...
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2answers
35 views

How to make closest possible to $n$ equidistant points in $\mathbb{R}^3$?

Given $n$ points in $\mathbb{R}^3$, the most exactly equidistant points we can have is $n=4$. I was wondering for larger cases how to construct the closest possible situation to having all points ...
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Intersection of a polar hyper plane and a quadric is a conic

Let $\mathbb{R} P^3$ be the real projective space of $\mathbb{R}^4$ and Q be a quadric defined as $$ Q = \left\{\,\left[\,\left(\begin{smallmatrix} x_1 \\ x_2 \\ x_3 \\ x_4\end{smallmatrix}\right)\,\...
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1answer
17 views

Need explanation of the spinor norm

Wikipedia and Groupprops gave a definition, but they didn't elaborate, so I don't understand, and there aren't cited sources on them. Is there any online sources that have proof on their basic ...
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1answer
19 views

Determine If Plotted Lines Segments Have Open Profile

Given the four line segments where $s= $start point $e = $end point. $- s(1,1)e(5,1) - s(5,1)e(5,5) - s(5,5)e(5,0) - s(5,0)e(1,0)$ The shape is drawn Which clearly has an open profile. Is there ...
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1answer
25 views

What is the Lemoine point useful for?

What is the Lemoine point useful for? Can someone give concrete examples /common example what math problems can be solved with usage of Lemoine point?
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1answer
32 views

Complex plane conundrum

So I've been wondering about geometry on the complex plane. Points on this plane are denoted by $(a+bi)$, right? If we have the point $(1+i)$, the horizontal distance from the complex axis is $1$, and ...
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22 views

Smallest box to fit cuboid in.

You have a cuboid with dimensions of 30x8x4 inside a box (cube). Every point of the cuboid touches some side of the box. What is the smallest box you can fit this cuboid in? I need the size of the box'...
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2answers
26 views

Trapezoid and line connecting two sides

Base sides of trapezoid are 8 and 6. Parallel line to base sides that connects two other sides splits trapezoid to two surfaces with equal area. What is the length of this line? My attempt: I tried ...
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1answer
18 views

Locus of points of the same distance from two diverging lines.

I am asked to find the locus of points of the same distance from two diverging lines in the the hyperbolic plane. I am using the Poincare model and am trying to use the unique common perpendicular to ...
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3answers
47 views

Efficient way to which is the point in between of three points [on hold]

I have three points $(x_1, y_1),~ (x_2, y_2),~ (x_3, y_3)$ that are on the same line. How to efficiently find which is the point in between. Example Also, is there any efficient way to check if 3 ...
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0answers
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Perspective plane projection based on 2 plane

I have two reference plane(same object) with different known height, captured from the same camera. Reference Plane 0, Reference Plane 1. The camera is tilted and the reference plane is flat. I am ...
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1answer
31 views

Does this definition of directional derivative depend on the magnitude?

Here is the definition: My calculus book defines directional derivatives for unit tangent vectors. According to Wikipedia, there is a convention that uses both the direction and magnitude. However, ...
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2answers
80 views

Prove a cyclic quad where there are parallel lines

In the following diagram, PT and PU are tangents. Prove that MUPT is a cyclic quadrilateral. In order to use the $\text{(ext $\angle = $ int opp $\angle$)}$ rule: $\widehat{U_4} = \widehat{...
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1answer
69 views

Finding the largest volume of a tetrahedron given the distances from the vertices to another point

There is a point $M$ in space. Four rays $MA$, $MB$, $MC$, and $MD$. Assume $MA=3$, $MB=4$, $MC=5$, $MD=6$. What is the maximum volume of tetrahedron ABCD? It is easy to show that M should be the ...
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1answer
24 views

Convert Covariance Matrix to Quadric Ellipsoid Form

Question: I have a 3x3 covariance matrix $M$ which I can plot in 3D to an ellipsoid (by eigendecomposition to get the principal axes and the scales). What I want is to project this covariance matrix ...
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1answer
23 views

Tracing a spheroid

If I have a spheroid $$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2} = 1$$ and I would like to draw a spiral (a helix) that starts from the bottom and traces its surface in spiral motion to the top. ...
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1answer
22 views

Confused with a spherical coordinate system surface element

I can not understand how a particular surface element is derived in spherical coordinates. The equation expressing the surface element vector is given as $$r_s = (\sqrt{R^2-z_s^2} \cos \phi,\sqrt{R^...
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0answers
23 views

What is the formula for graphing a circular base for either cones or cylinders? [on hold]

Assuming that I've already plotted a cone or cylinder, what is the formula to create the base of the shape?
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2answers
28 views

Component-Wise 3D Vector Projection

I want to define the projection of a vector $\mathbf{v} \in \mathbb{R}^3$ onto the line $\mathbf{r} \in \mathbb{R}^3$ in terms of the components of $\mathbf{v} = (v_x, v_y, v_z)$. In 2D, this looks ...
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1answer
19 views

Simson line problem with understanding

Lines $k, l, m$ intersect in one point $O$, and point $P$ doesn't lie in any of them. Points $A, B, C$ are orthographic projection of point $P$ on lines $k, l, m$ Prove that orthographic projection $P$...
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2answers
42 views

Parallelogram of maximal area inscribed in a triangle [on hold]

Let $ABC$ be a triangle. Find the points $D$, $E$ and $F$ lying on $[AB]$, $[BC]$ and $[AC]$ such that $ADEF$ is a parallelogram with maximal area. NB: I am in 9th grade. The teacher said to me ...
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2answers
49 views

Finding the angle between the lines represented by $9x^2+24xy+16y^2=0$

Question: The angle between the lines represented by $$9x^2+24xy+16y^2=0$$ is (A) $90^\circ$ (B) $0^\circ$ (C) $180^\circ$ (D) None of Above I’ve tried to solve it by making ...
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0answers
30 views

Representation of fundamental form by positive function

I want to solve the following exercise: Let $\omega = \frac{i}{2\pi} \sum dz_i \wedge d\overline{z_i}$ be the standard fundamental form on $\mathbb{C}^n$. Show that one can write $\omega = \frac{i}{...
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1answer
35 views

Finding Cartesian coordinates of remaining vertices of triangle, given angle from y-axis with point A and a vertex [on hold]

I have an isosceles triangle $ABC$, where the height $h$ and angle at point $A$ are known. The Cartesian coordinates of point $A$ are also known. How to find $B = (x_{2},y_{2})$ and $C = (x_{3},y_{3})$...
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Solid angle: Must a region subtending a solid angle be (simply) connected?

Although answers to the question "What is a Solid Angle?" explain that the shape of the area subtending a solid angle doesn't matter, my question is if the region has to be simply connected (no holes)....
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0answers
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Quantifying positining and alignement of two line segments

For simplicity we can consider a 2D case: where we have line segments of the same length $l,$ and we suppose we have $n$ of them with their centre coordinates and orientations randomly assigned. For ...
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1answer
22 views

Rotate a Point on an ellipse by an angle and calculate the distance between them

I am given a starting point $S=(s_1,s_2)^T$, the Center Point $C=(c_1,c_2)^T$ of the ellipse, with major radius $a$ and minor radius $b$, also the major axis is rotated w.r.t. the X axis by an angle ...
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1answer
34 views

Limit question concerning a triangle and its circumcircle

$ABC$ is an isosceles triangle ($|AB|=|BC|$). Let $s$ be the length of the altitude from vertex $B$ to side $AC$, and let $m=|AC|$. Given that the radius of the circumcircle of $ABC$ is $2\,\text{cm}$,...
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2answers
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How can line segments that don't meet be called “perpendicular”?

Suppose there is a line segment from $(4,0)$ to $(6,0)$, and another line segment $(0,1)$ to $(0,2)$. They don't form an angle, so how are they "perpendicular"? What actually is meaning of it? Like ...
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2answers
44 views

To find a locus of a point P inside triangle ABC

ABC is a triangle. P is a point inside ABC such that its distances from the sides of Triangle ABC are x, y, z. If a, b, c and k are given set of constants, prove that the locus of P such that ax+by+cz=...
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1answer
30 views

Prove that, reflection of vertex of a triangle about angle bisector through other vertex lie on opposite side of triangle.

$A(1,3)$ and $C({-2\over5},{-2\over5})$ are the vertices of a triangle $ABC$ and equation of angle bisector of $\angle ABC$ is $x+y=2$. Find equation of side $BC$ I tried a lot to solve this question,...
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2answers
48 views

Prove $\triangle$ AB$\Omega$ $\cong$ $\triangle$CD$\Lambda$.

In omega triangle $\triangle AB\Omega$ and $\triangle CD\Lambda$, given $$\angle A \cong \angle B\,,~~\angle C\cong \angle D\,,~~\text{and}~~ \overline{AB} \cong \overline{CD}~,$$ prove $\triangle AB\...
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0answers
25 views

Another approach to Soddy circles formula

I am trying to prove a version of Descartes' theorem in an elementary way. Given three mutually tangent circles (no one is inside the others) with centres $A,B,C$ and radii $a, b, c$ respectively, I ...
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0answers
20 views

Determine the ratio between the radii lengths of the circumcircle and inscribed circle

Find the ratio between the radii lengths of the circumcircle and the inscribed circle of a triangle I have tried using the rule: $$\frac{a}{\sin{A}}=\frac{b}{\sin{B}}=\frac{c}{\sin{C}}=2R$$ Where R is ...
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1answer
44 views

How to find the number and coordinates of self-intersections points for a polygon?

I have a self-intersecting polygon defined by $n$ points on the plane: ...
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1answer
51 views

Simple Geometric Proof of Intersecting Circles

The following problem has been bothering quite a while. I guess there is a gap in my school knowledge of geometry, but I do not know how to show that: Prove or disprove that given two circles with ...
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2answers
44 views

For which $\alpha$ will take the cake ever be again with chocolate on the bottom and cream on the top

Question: A bored kid left alone at home decides to take a chocolate cream cake (chocolate on the bottom, cream on top) and his protractor and spend the day as follows: He cuts a slice of angle $\...
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2answers
28 views

Probability on a disc

I was solving some questions of probability and I came across the following one: Question: Given an arbitrary disc with radius $r> 0$. A point is chosen randomly on the disk. Determine the ...
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1answer
52 views

How does one find a circle which intersects only once with three other circles?

Say I have the equations of three different circles on a plane. How would I proceed to create a fourth circle, which intersects only once with each of these known circles? Would that be possible at ...
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1answer
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How is the frenet frame along an asymptotic curve related to the geometry of the surface?

I'm reading Differential Geometry: A first course in curves and surfaces by Theodore Shifrin and here is one of the questions from the exercise. I just can't seem to make the connection between the ...
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0answers
22 views

Representing a complex line as a directed ellipse

Consider nonzero $v = v_r + iv_i \in \mathbb{C}^n$, It can be thought of as an ordered 2-tuple of vectors $(v_r, v_i)\in \mathbb{R}^n\times\mathbb{R}^n$. The complex line generated by $v$ is $$\{r[(\...
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0answers
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How to find the second “polar” coordinates?

How can I find the coordinates of A,B respective to the centre of pin 1? So far I have used used $6.5\sin45$ to find $X,Y$ from pin $1$ and tried the same trick to find $A,B$ with $1.25\sin45$ deduce ...
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2answers
38 views

Conrad's $\mathit{Dihedral\ groups}$: Rigid motions taking a regular $n$-gon back to itself carry vertices to vertices

I have been reading Keith Conrad's expository paper Dihedral groups I and I have two questions about Theorem $2.2$, which deals with the size of $D_n$. In the first part of the proof you can read ...
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0answers
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Are two circles of different diameters similar on a sphere? [on hold]

The teacher told me that there are only congruent triangles on the sphere, but no similar triangles. My question is, are the two circles with different diameters on the sphere similar? I'm asking a ...
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2answers
58 views

what is the apparent size of a 1' object that is 600' away?

I am trying to impress US Coast Guard search and rescue experts how small the human head is (say, 1' in size) when the person waving to us is 600' away (roughly 1/10th of a nautical mile.) When I ...