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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Finding area of a triangle from incircle of a square and given intersection lines.

The incircle of square $ABCD$ touches sides $\overline{AB},$ $\overline{BC},$ $\overline{CD},$ and $\overline{DA}$ at $P,$ $Q,$ $R,$ and $S,$ respectively. Point $E$ lies on arc $PQ$ of the incircle. ...
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Getting a Bivariate Polynomial From a Zero-Set

Given the zero-set $\{x_0\}\subset\mathbb R$ of a function of the form$$ax+b,$$I can find one such function via$$f(x)=x-x_0;$$given the zero-set $\{x_0,x_1\}\subset\mathbb R$ of a function of the form$...
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Volume of truncated pyramid triangle base

Find the volume of a truncated pyramid with height 10. One base has sides 27, 29, and 52. The truncation base has a base perimeter of 72. I understand this is a pyramid minus a pyramid question but ...
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Prove that orthocenter of the triangle formed by the arc midpoints of triangle ABC is the incenter of ABC

Let ABC be an acute triangle inscribed in circle W. Let X be the midpoint of the arc BC not containing A and define Y, Z similarly. Show that the orthocenter of XYZ is the incenter I of ABC This is ...
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Calculus: Early Transcendentals, 7th ed(stewart)-chapter 12 problems plus exercise 3

I tried to resolve the following excercise but i got stuck. 3)Let be $L$ the line of intersection of the planes $cx+y+z=c$ and $x-cy+cz=-1$, where $c$ is a real number. (a) Find symmetric equations ...
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For angles $A$ and $B$ in a triangle, is $\cos\frac B2-\cos \frac A2=\cos B-\cos A$ enough to conclude that $A=B$?

Brief enquiry: $$\cos\frac B2-\cos \frac A2=\cos B-\cos A$$ Optionally $$\sqrt\frac{1+\cos B}{2}-\cos B=\sqrt\frac{1+\cos A}{2}-\cos A$$ Is above equality sufficient to prove that it ...
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Solving for side lengths of a triangle given specific conditions.

For a given constant $b > 10,$ there are two possible triangles $ABC$ satisfying $AB = 10,$ $AC = b,$ and $\sin B = \frac{3}{5}.$ Find the positive difference between the lengths of side $\overline{...
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Find radius of largest circle within ellipse $\frac{x^2}{9} + \frac{y^2}4 = 1$ with their intersection only at $(3,0)$

An ellipse is defined by the equation $$\frac{x^2}{9} + \frac{y^2}4 = 1$$ Compute the radius of the largest circle that is internally tangent to the ellipse at $(3,0),$ and intersects the ellipse ...
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Maximum social distancing puzzle

A spherical planet of radius $R$ is inhabited by $n\ge 2$ aliens. Due to the emergence of a novel deadly virus, the aliens must practice social distancing, staying away as far as possible of each ...
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Parametric curve of a spiral with straight and semicircular parts

I need to design a fully parameterized spiral like this where I can choose/change the central part length and the radius of the innermost curve. I understand that it will be somehow related to the ...
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1answer
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Test if a point on a hexagonal lattice falls on a specified superlattice?

Based on previous answers (1, 2, 3) integers $i, j$ produce a hexagonal lattice using $$x = i + j/2$$ $$y = j \sqrt{3} / 2.$$ From a point $k, l$ I can make a superlattice from integers $I, J$ ...
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Look at equation $6xy - 30x + 20y - 100 = 0$. Do transformation coordinate axes to change the cone section to standard form!

The full question: Look at equation $6xy - 30x + 20y - 100 = 0$. Do transformation coordinate axes to change the cone section to standard form? Sketch out the cone sections graph Its my task from my ...
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Conjugate diameters of ellipsoid

I am familiar with ellipses,but am seeking to find the condition for 2 diameters of an ellipsoid to be conjugate.Any help or references would be appreciated.
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Obtaining condition for existence of two Chords through a single point with their Mid-points on the $x$-Axis

Given a circle $x^2+y^2=\alpha x + \beta y$, obtain the condition on $\alpha$ and $\beta$, if two distinct chords can be drawn through the point $(\alpha, \beta)$ such that their mid-points lie on the ...
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1answer
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Is there a geometric proof for distributivity of integer addition/multiplication and other similar properties?

I am aware that commutativity, associativity and distributivity of integer addition and multiplication follow from their standard set theoretic definitions but I am looking for something suitable for ...
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In the case of symmetric matrix diagonalization, why we take orthonormal eigenvector matrix?

For a matrix $A$ we can use diagonalization using this formula: $$ A = X D X^{-1}$$ where $X$ is a matrix containing eigenvectors in the columns and $D$ is the diagonal matrix containing eigenvalues. ...
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Finding the area of a fraction of a circle defined by points on a grid.

A grid is shown below, where the shortest distance between any two points is 1. Find the area of the circle that passes through points $X,$ $Y,$ and $Z.$ Can it be assumed that the arc created over ...
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Hint as to how to formalize that the curvature of a curve inside the unit circle is bounded below by the curvature of the circle

I am trying to self teach differential geometry and to that effect I am trying to do the Homework in the MIT open course. The specific question I am struggling with is: Let $c$ be a regular curve ...
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Are the two triangles similar?

A book I was answering was asking for the center of enlargement and the scale factor of the two similar figures.But I don't think the two triangles are similar. They corresponding sides don't have the ...
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Convert linear distance to steering angle

I need to calculate the angle of the front steering wheel using a collapsible piston(linear sensor). 'x' is used to represent the length in inches of the movable part of the sensor and is the ...
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Chord of contact to a second degree homogeneous curve [closed]

How can I prove that equation of chord of contact to a second degree homogeneous curve of the form of ax² + by² + 2hxy + 2gx + 2fy + c is T = 0 also that equation of a chord with a given middle point ...
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1answer
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Right circular cone shortest distance question from a junior high school st. [duplicate]

Ant's shortest distance that it can travel Can you show me the answer in the cone unfolded. I would be glad if you guys can show me the solution in a simple way because I am a high school student.
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What logic can I use to determine the complete overlap of multiple polygons?

I need to measure the 2D space that multiple arbitrary polygons occupy. To do so, I need to add the area of all polygons and subtract the overlap. I already have written a function that takes in only ...
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Are there geometries in which points have finitely many parallel lines?

In Euclidian geometry the Parallel Postulate holds that given a parallel line as a reference there is one parallel line through any given point. In elliptic geometry there are no parallel lines. In ...
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Distance between two generated lines

Write the equations of the rectilinear generatrices of the hyperbolic paraboloid $\frac{x^2}{p}-\frac{y^2}{q}=2z\ \ ,p,q>0$. Out of these rectilinear generatrices select those which are parallel to ...
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Intersection of oblate spheroid $\frac{x^2+y^2}{R_e^2}+\frac{z^2}{R_p^2} =1$ and plane $n_xx+n_yy+n_zz=0$

To calculate the distance between two points on Earth, I used 3 different approaches. For small distances, I used the Euclidean distance. For medium distances, I used the arc length on the circle ...
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1answer
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When does a convex set have a unique outward normal direction?

Let $C$ be a closed, nonempty, and convex set (in a real Hilbert space $\mathcal{X}$), and let $c\in C$ be a point on its boundary. When will the normal cone $N_Cc$ have a unique (nonzero) direction? ...
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3answers
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Finding fraction of a volume by visualization of cube being cut into separate pieces

Cube $ABCDEFGH$ is cut into four pieces by cutting along planes $BCHE$ and $BDHF.$ Find the fraction of the volume occupied by the piece containing the vertex $A.$ I'm not really able to visualize ...
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6answers
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Finding triangle side lengths given altitudes or lengths of sectioned triangles within triangle.

In right triangle $ABC,$ $\angle C = 90^\circ.$ Let $P$ and $Q$ be points on $\overline{AC}$ so that $AP = PQ = QC.$ If $QB = 67$ and $PB = 76,$ find $AB.$ How do I use ratios and given side lengths ...
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Finding the radius of a circle given lengths between different points within the circle.

The diameters $\overline{WX}$ and $\overline{YZ}$ of a circle intersect at the center $O$ at right angles. A smaller circle is tangent to the first circle at $X,$ and intersects $\overline{WO}$ and $\...
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25 views

Finding a chord length from other given chord lengths.

Let $E,$ $F,$ $G,$ and $H$ be points on a circle such that $EF = 22$ and $GH = 81.$ Point $P$ is on segment $\overline{EF}$ with $EP = 12,$ and $Q$ is on segment $\overline{GH}$ with $GQ = 6.$ Also, $...
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60 views

Complex numbers on quadrilateral in cyclic order

Let $z_1, z_2, z_3, z_4 ∈$ $\mathbb{C}$ such that these numbers are in quadrilateral appear in a cyclic and positive order, prove that these numbers are contained in a circumference if and only if: $$...
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Randomizing a regular polygon along its “spokes” while maintaining area

I have a 2d graphics question that seems like it'd fit better here than at stackoverflow. Please forgive any breach of ettiquette as I am new to mathematics exchange. I did search for a solution ...
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1answer
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Finding the diagonal of a rectangle.

What fraction of the rectangle is shaded? (You may assume that each line, other than the diagonal of the rectangle, is parallel to some side of the rectangle.) Is there a way to solve this without ...
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1answer
19 views

Sine difference identity

I'm trying to prove that given 2 vectors $\vec{a} = A(\cos{\alpha}, \sin{\alpha}) $ and $\vec{b} = B(\cos{\beta}, \sin{\beta}) $ the following relation is true by using the exterior product with the ...
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1answer
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Found the inscribed between an arc and its chord circles radii ratio given that the chord intersects with the inner circle diameter at $30^\circ$

I found the problem here (I can't see deleted posts) but the post got downvoted and deleted soon, but I felt so inspired finding the solution that can't let the problem rot by itself, so, rather re-...
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36 views

Isosceles trapezoid and cyclic quadrilaterals [closed]

Let $ABCD$ and isosceles trapezoid . Let $A_1$ be an arbitrary point on $AC$ , the circumcircle of $\triangle A_1CD$ intersects $BC$ at $B_1$ .Let $M$ and $N$ be the midpoints of $AC$ and $BC$ ...
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Can a curve of constant width have a straight side?

It seems that the curves of constant width that I've seen are all "curves" in the sense that any support line only cut through one point. I just wonder if it could have a straight side?
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Compute connection matrix

The (pseudo)Riemannian metric and the connection in $R^3$ with coordinates $x=u_1$,$y=u_2$,$z=u_3$ are given on the basic vector fields by $$(\partial_{u_i}, \partial_{u_j})=\frac{\partial^3f}{\...
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1answer
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sector area of a circle [closed]

A little circle and big circle share the same center. The radius of the little circle is 4 cm and the radius of the big circle is 9 cm. What is the area of the shaded region shown if the measure of ...
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1answer
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Find the shaded area, given the sides:

The given sides were $AC=BD=25$, $AD=BC=15$, and $DC=7$. No other explanations were given to this problem. I tried to connect A and B to form an isosceles trapezoid then tried to work around with ...
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Can a polygon be monotone with respect to precisely one direction?

This question is first exercise in section 2.2.3 in book "Computational Geometry in C". I want to check if such a polygon exists, and if it does, then prove that it is indeed monotone with respect to ...
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Question from PRMO 2019.

An ant leaves the anthill for its morning exercise. It walks 4 feet east and then makes a 160° turn to the right and walks 4 more feet. It then makes another 160° turn to the right and walks 4 more ...
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29 views

Can I make this assumption?

I am solving this question: Let $ABC$ be an acute angled triangle and $CD$ be the altitude through $C$. If $AB=8$ and $CD=6$, find the distance between the midpoints of BC and AD. So I observed that ...
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2answers
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Finding length of line that intersects trapezoid diagonals.

In trapezoid $ABCD,$ base $\overline{AB}$ has length 6, and base $\overline{CD}$ has length 18. A line passes through the intersection of the diagonals, parallel to the bases. This line intersects $\...
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4answers
47 views

Find the area of a rectangle in a unit square.

In the unit square $ABCD,$ $P$ is the midpoint of $\overline{AB},$ and $PQRS$ is a rectangle. Find the area of rectangle $PQRS.$ I'm not quite sure how to go about solving this problem. I assumed ...
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2answers
36 views

Finding the hypotenuse of a triangle using angles and segment lengths. [closed]

In the diagram, $\angle CAB = 90^\circ.$ Let $D$ be a point on $\overline{AB},$ and let $E$ be a point on $\overline{AC},$ such that $AB = AC = DE,$ $BD = 9,$ and $CE = 8.$ Find $DE.$ [asy] unitsize(...
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Understanding SE(3) Twists

I am just learning about twists to represent 3D velocities, and I have two questions: 1) Wikipedia defines a twist as "an angular velocity around an axis and a linear velocity along this axis". To ...
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3answers
47 views

Form a new base with two vectors

I need to make a new orthonormal base in $\mathbb{R^3}$ given $(2,7,5)$ and $(4,1,3)$ so that it makes $(\widehat{e_1}, \widehat{e_2}, \widehat{e_3} )$. But $ \widehat{e_1} $ has the same direction of ...
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Prove that a torus triangulation cannot have degrees of vertices $5, 7, 6, 6, 6, 6, \ldots$

I found one rather interesting but intractable topology problem. Prove that a torus triangulation cannot have degrees of vertices $5, 7, 6, 6, 6, 6, \ldots$ Despite various attempts to contract ...

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