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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Angles of an incircle

In triangle $ABC$, $\angle BAC = 72^\circ$. The incircle of triangle $ABC$ touches sides $BC$, $AC$, and $AB$ at $D$, $E$, and $F$, respectively. Find $\angle EDF$, in degrees. I am currently taking ...
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Geometry vectorial

Hello i have a question, what's is the vector of this equation? Find the vectorial equation of this exercise. Contain the point (7, 3, 1) and intersect perpendicular to the rect (x, y, z) = (1, 2, 0)+ ...
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Integrating over a hypersphere surface

I have an $n$ dimensional hypersphere $S$ with radius $D$. Along one axis, I cut off the caps of the sphere at a position of $-a$ and $a$. I wish to find the remaining surface area of the hypersphere. ...
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Regarding an isometric drawing of a circle, how does this align perfectly?

In the above image, the circle is drawn as an isometric view. Why does it align well when we draw a six-pointed star? The radius of the larger arc is thrice the radius of the smaller arc. My question ...
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How can curves approximate curves, but lines cannot approximate lines?

This is my first question on StackExchange, and i have very little knowledge of mathematics to tell if it is a physics issue or a math issue, so any info will be useful. Main question Let's say i want ...
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How to rotate a vector to find its location in the image that has been rolled

I have a camera with the center of elevation/pitch:87.3837 roll:-0.5763 yaw/azimuth:-35.8876 and object with ...
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Circle problem intersecting

Two circles $C_1$ and $C_2$ pass through the centres of each other and intersect at $X$ and $Y$. Chord $YA$ of $C_1$ intersects $C_2$ again at $B$. If $AB= a$, radii $C_1$ and $C_2$ are $r_1$ and $r_2$...
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Conic sections appendix [duplicate]

My questions concerns a chapter in Michael Spivak's math book Calculus. On page 81, Spivak talks about finding a coordinate axes for a plane P that intersects a cone. What I don't understand is the ...
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60 views

Is this line uniquely determined?

Let $\alpha: x-y+2z-2=0$ and $\beta: x-2y-2z+3=0$ be two planes in $\mathbb{R}^3$. I am asked to find a line $d_1\subset \alpha$ such that $d_1$ is perpendicular on the line $\alpha \cap \beta$ and ...
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Conditions for existence of $k$ vectors In $\mathbb R^k$ with given pairwise angles $\theta_{ij}$

Given angles $0<\theta_{ij}<\pi$ for $1\leq i<j\leq k,$ what conditions are there on the angles to ensure that there exists $k$ unit vector $v_i\in \mathbb R^k$ so that the angle between $v_i$...
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Get a rotation matrix with euler angles without rotation matrix per axis

From this question, I managed to rotate vectors without euler angle rotation matrix. But I'm still struggling with the case which needs euler angles. I'm already aware of the rotation matrix something ...
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Geometric reason why this determinant can be factored to (x-y)(y-z)(z-x)?

The determinant $\begin{vmatrix} 1 & 1 &1 \\ x & y & z \\ x^2 & y^2 &z^2 \\ \end{vmatrix}$ can be factored to the form $(x-y)(y-z)(z-x)$ Proof: Subtracting ...
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Proof of equation resembling power of a point in triangle

I've tried this problem for a while now, and I'm stuck on it. Let $D$ be the foot of altitude $AD$ on side $BC$ of triangle $ABC$. Denote a point $N$ on altitude $AD$. Prove that this point $N$ is the ...
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Circle under tangents meets incircle at same point on BC

Inside $\triangle ABC$, let a circle $\omega$ be tangent to sides AB and AC, not touching BC. Tangents from B and from C to $\omega$ (different from the triangle sides) intersect at point X. Show that ...
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38 views

Find the area of rhombus $ABCD$

This is a problem for homework that I'm stuck on. [![enter image description here][1]][1] So far, I've only drawn the line $GA$ and found that it was $12$ but I haven't been able to find anything else....
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How to generate random velocity vectors that can only move an object forward within a valid arc?

I have an object with known coordinates in in 3D but on the ground (z=0). The object has a direction vector. My goal is to move this object on the ground (so ...
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Finding the ratio of the area of two right triangles $HCD$ and $ABD$

In the right triangle $ABC$, we have $AB=\sqrt3$ and $AC=2$. what is the ratio of the area of two right triangles $HCD$ and $ABD$ ? $1)\dfrac37\qquad\qquad2)\dfrac47\qquad\qquad3)\dfrac{16}{21}\qquad\...
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Finding cyclic quadrilaterals after drawing orthic triangle of a triangle

Find the six cyclic quadrilateral's in the figure Source: Euclidean Geometry in mathematical Olympiads,page-7 I found three already : AEFH,BFHD, EHCD by the property the if opposite angle of a ...
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How to find the equation of a line and of a plane in $\mathbb{P}^3(\mathbb{R})$?

I have the following points in $\mathbb{P}^3(\mathbb{R})$: $A=[1, 0, 1, 1]$, $B=[1, 1, 0, -1]$, $C=[1, 1, 1, 1]$, $D=[0, 1, 0, 1]$, $E=[-1, 2, 3, 2]$. I am asked to find the intersection of the line $...
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35 views

Some questions involving two lines and two planes in $\mathbb{R}^3$

Let $d_1:\begin{cases} 2x+y-z=1 \\ x-z=2\end{cases}$ and $d_2:\begin{cases} x-y+2z=1 \\ x-y=2\end{cases}$ be two lines in $\mathbb{R}^3$. I have to find the plane $\pi_1$ that contains $d_1$ and is ...
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Proving that a square is made by connecting point-opposite midpoint in larger square

Below is a diagram of a square, where $E, H, F,$ and $G$ are the midpoints of the square. I want to prove that the smaller square formed by the intersections of $EC, FD, BH,$ and $AG$ is a square. So ...
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Surface area and volume for Minkowski sum

Let $K_1$ and $K_2$ two convex bodies in the euclideand space $\mathbb{E}^3$ (of dimesion 3): Let's denote the $V$ as the volume and $S$ the surface area I am looking for two thing(with reference): 1-...
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Construction of angle bisector of a given angle

Steps of Construction : Taking B as centre and any radius, draw an arc to intersect the rays BA and BC, say at E and D respectively [see Fig.11.1(i)]. Next, taking D and E as centres and with the ...
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Euclid's fourth postulate and cone points

In the following answer answer , Euclid's right-angle postulate excludes the existence of cone points: right angles at the vertex of a cone are smaller than right angles elsewhere on the cone. So ...
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42 views

If you know three angles in a triangle are equal (meaning each measure 60 degrees), does that mean the triangle is equilateral?

If you know three angles in a triangle are equal (meaning each measure 60 degrees), does that mean the triangle is equilateral? Meaning, if we know the three angles are equal, does that mean the sides ...
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Minimum cost required to cover a line with line segments.

We are given a line segment $[1, n]$ with $m$ smaller line segments $[l_{i}, r_{i}]$. An example being $[1, 4]$ and line segments ${[1,2], [2,3], [3,4]}$. We can cover this using first and third ...
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Determinant not equal to volume error (closed)

The determinant of a $3\times 3$ matrix $\begin{vmatrix} 1 & 1 &1 \\ x & y & z \\ x^2 & y^2 &z^2 \\ \end{vmatrix} $ is the volume of a parallelopiped with its three sides as ...
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102 views

Angles in $\mathbb{R}^{n}$

Given a vector $v\neq0$ in the plane $\mathbb{R}^{2}$, we know that the angles $\theta_{1}$ and $\theta_{2}$ between $v$ and the $x$-axis and the $y$-axis respectively, sum to $\frac{\pi}{2}$; $$\...
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Secondary school circle geometry [closed]

I have a question posed to me. I experience difficulty attempting part (a) on its own. On the other hand, I can use part (c) to show (a). Kindly advise. Thank you.
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Find equation for a parabolic line that goes through two points in 3D space

I'm currently working on some computer graphics program where I want to show a line that "bounces" from point $A$ to point $B$, so I need an equation through which I can plug in values and ...
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Find value of angles of the inscribed triangles in a regular hexagon

^ Image of the diagram The diagram involves 6 evenly-spaced out points on a circle, and three different types of triangles inscribes in it using the points as vertices (I have drawn scalene, ...
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Defect of two triangles inside a Saccheri quadrilateral

Let $\square ABCD$ be a Saccheri quadrilateral with base $\overline{AB}$. Let $E$ be the point on $\overline{AB}$ such that $\square EBCD$ is a Lambert quadrilateral with acute angle $D$. Is it true ...
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Using pole and polar to prove perpendicularity in a triangle with the circumcenter

Given triangle $ABC$ with the circumcircle $(O)$. An arbitrary line meets the sides $AC$, $AB$ at $D$, $E$, respectively, and meets $(O)$ at $P$ and $Q$. Let $BD$ meets $(O)$ at $M$, $CE$ meets $(O)$ ...
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In a six point circle, how many different types of triangles can be formed using the points as vertices?

Attached is the image I have drawn so far for this question. https://imgur.com/a/En3oH3b I got three. Are there any other different types of triangles that can be drawn? I also have another question. ...
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71 views

Angle bisectors of triangles

In triangle $ABC,$ angle bisectors $\overline{AD},$ $\overline{BE},$ and $\overline{CF}$ meet at $I.$ If $DI = 3,$ $BD = 4,$ and $BI = 5,$ then compute the area of triangle $ABC.$ We are learning ...
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Given a direction, in vector form, find the closest point on a circle with a given normal plane. [closed]

I have a direction in vector form that I am using to find the closest point on a circle. The circle exists in a 3d space. I have the center of the circle in form {x, y, z} and the radius. I got this ...
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Solving A System of Two Inequalities Each With Two Variables

I am writing an image processing algorithm. The algorithm calculates a random contrast adjustment, and a random brightness adjustment, and applies those to each pixel in an image, like... ...
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53 views

Area of a triangle inside a square

$ABCD$ is a square and $S_{ABCD}=2016$. $M$ is the midpoint of $AB$ and $O=AC\cap BD,$ $N=BD \cap CM$ and $P=AC\cap DM.$ Find the area of $MNOP$. So we can write the area of $MNOP$ as $$S_{MNOP}=S_{\...
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Can an ellipse inscribed in a circle have a constant ratio of chords?

This is a problem I've been trying to solve through simple algebra but the calculations get way too complicated so I thought I'd ask for some help. Assume an ellipse inscribed in a circle centered at $...
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What does orthogonality between two vectors really represent?

The Pythagorean Theorem states that: $$\|c\|^2=\|a\|^2+\|b\|^2 $$ Now because in general two vectors $u$ and $v$ are orthogonal if and only if: $$\|u-v\|^2=\|u\|^2+\|v\|^2 $$ If the sum of the squared ...
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Check if a point lies within two standard deviations of a multivariate Gaussian?

Assume that I have a multivariate Gaussian distribution $\mathcal{N}(x;\mu,\Sigma)$ in $D$-dimensional space parameterized by a mean vector $\mu \in \mathbb{R}^D$ and a covariance matrix $\Sigma \in \...
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1answer
68 views

Geometry Question about a triangle

I was sent this by a friend who could not solve it and I did the problem, and got the answer of 3/8 using similar triangles and then using areas perpendicular heights and bases. Can someone please ...
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39 views

Trying to create a 3D model of Kelvin’s Tetradecahedron / Tetrakaidecahedron polyhedra

How can I go about creating a 3D model / 3D image of a Kelvin’s Tetrakaidecahedron Cell / Tetrakaidecahedron. I planned on using Octave to 3D model an image it mathematically then convert that into a ...
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68 views

Minimum area binding box for set of points [closed]

I am writing some paper and stuck at proving that in min area binding box there needs to be 2 points on the edge of min rectangle. Every single time I make a new case I got that. By getting extreme ...
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76 views

Show that $\sin^2c = \sin^2a + \sin^2b$. True?? [closed]

The vertex $B$ of the triangle $ABC$ lies in the plane $P$. The plane of the triangle meets the plane in a line $L$. The angle between $L$ and $AB$ is a, and the angle between $L$ and $BC$ is $b$. The ...
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1answer
51 views

Prove that a set is the convex hull of another set

Consider a set $\mathcal{P}$ defined as $$ \mathcal{P} := \left\{ (x,y) \mid x\in[0,\bar{x}], y\in[0,\bar{y}], xy=0 \right\}.$$ I am trying to prove that $\operatorname{conv}(\mathcal{P}) = \tilde{\...
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49 views

A simple Geometry Doubt

In the figure below there are two congruent squares, then what fraction of larger circle area is the green circle area? Basically i assumed that the points of tangencies of green circle and the ...
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31 views

Construct line segment $XY$, point $M$ is inside a square, points $X, Y$ lie on the sides of the square, $|XM| = |YM|$

Construct line segment $XY$, point $M$ is anywhere inside the area of an arbitrary square, points $X, Y$ lie on the sides of the same square, $|XM| = |YM|$. Allowed tools: ruler, compass. No ...
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Having trouble with the following geometry problem [closed]

In Triangle $ABC$, points $P$ and $Q$ lie on sides $AB$ and $AC$, respectively, such that the circumcircle of Triangle $APQ$ is tangent to side $BC$ at $D$. Let $E$ lie on side $BC$ such that $BD\cong ...
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31 views

Triangle of least perimeter for a given area is the equilateral. [duplicate]

I want the opposite of this question, i.e., a proof that the equilateral triangle has the least perimeter of all the triangles with a given area. If possible I'd prefer an answer without Lagrange ...

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