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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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Dividing Hypercubes into $n$ smaller Hypercubes

Name a positive integer $n$ nice if a square can be divided into $n$ smaller squares. The smaller squares do not need to be of the same size. Since you can always divide a square into $4$ smaller ...
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22 views

Algorithm to draw an image within arbitrary size rectangle

I'm writing an application that needs to draw an arbitrary size image (width and height are random at the input) within another randomly generated rectangle(dimensions given as input) so that the ...
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How can I find the top and bottom most point from the intersection of three circles?

I have three circles with equal radius ($r$). I want to formulate the bottom and topmost point of their intersection as a function of $r$.
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Prove that the straight lines whose direction cosines are given by the relations

Prove that the straight lines whose direction cosines are given by the relations $al+bm+cn=0$ and $fmn+gnl+hlm=0$ are parallel if $\sqrt {af} \pm \sqrt {bg} \pm \sqrt {ch}=0$. My Attempt: Given: $$...
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square proof using intersection of mid points [on hold]

In square $ABCD$, $E$ is the midpoint of $\overline{BC}$, and $F$ is the midpoint of $\overline{CD}$. Let $G$ be the intersection of $\overline{AE}$ and $\overline{BF}$. Prove that $DG = AB$.
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Describe the shape of projection of vertices (vector positions of a cube) onto a 2D plane from a source (position vector)?

I am having trouble with this. I can manually calculate every single projection point onto the z=0 plane from deriving vector equations to get to the z plane for each vertices. From this I can then ...
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1answer
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Conjecture: If $A^\prime$ is outside the circumcircle of $\triangle ABC$, then $\triangle A^\prime BC$ has a larger circumradius

While solving a few problems, I came across a property of triangle. It looks simple, but I am not able to prove it. Simply stating it: Conjecture. Given $\triangle ABC$ and its circumcircle, ...
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3answers
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Right cone, you are at A and need to complete a revolution before reaching the bottom B. What is shortest distance AB?

You are on a mountain that is a right cone shape. You are trying to get to B and you are somewhere up the mountain A such that you lie on the line OB. The line AB must do one full revolution of the ...
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Introduction to Geometry books

I am looking for a book that covers introduction to geometry. Currently, I am reading Geometry: A Metric Approach with Models, by Richard Millman. However, I noticed that most theorems and all of the ...
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1answer
33 views

Perpendicular bisector in a different metric

Consider two points $a, b \in \mathbb{R}^2$. Then from elementary geometry, the set of points that are equidistant from both $a$ and $b$ is precisely the perpendicular bisector of the line segment $ab$...
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How can I solve this word problem? [on hold]

I was given this word problem to solve: The base of a ladder is 8 feet away from the edge of a building. The ladder is 17 feet long. How high up on the building does the ladder reach? I need ...
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3answers
52 views

what geomatric series formula is used?

trying to follow a solution related to geometric series, but not sure what formula is used here. Any pointer is appreciated.Image here, can't embed image yet. I do try to plug in the Sn formula, but ...
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snub cube angles

I am trying to build a snub cube. I have made 6 squares and 32 equilateral triangles (out of perler beads if you're curious). I am trying to figure out the angles at which I adjoin the squares to the ...
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0answers
15 views

Products of subgroups of Euclidean Group

So, I'm reading a book and I need help with some stuff. The book defines product of subgroups $G_1, G_2$ of $G = E(n)$ as $G_1G_2 = \{g_1g_2|g_1 \in G_1, g_2 \in G_2\}$, which is not necessarily a ...
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1answer
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AD and BE are the altitudes of the triangle ABC with orthocentre H,which lies in the interior of the triangle.If BH=AC,Find angle B

After forming some equations we get angle DAC=EBC .After this I hit a dead end but I think this would require cosine law or some other trignometric relation.Please help (HAVE BEEN WORKING ON THIS FOR ...
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1answer
43 views

Average distance between point in a disc and line segment

What is the average distance between a (randomly chosen) point in a disc of radius r and a line segment of length $a < 2r$ whose midpoint is at the center of the disc? ["Distance" here being the ...
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2answers
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If $12$ distinct points are placed on a circle and all the chords connecting these points are drawn, at how many points do the chords intersect?

If $12$ distinct points are placed on the circumference of a circle and all the chords connecting these points are drawn, at how many points do the chords intersect? Assume that no three chords ...
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2answers
33 views

bijective function from a circle to a line

I am just reading a book which says the square has the same size as the line. It is saying every coordinate in the square have two coordinates x and y, and we can make a bijective function this in a ...
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1answer
31 views

Is it possible to identify a circular arc given its length and two of its endpoints?

Suppose you have the length of the circular arc AB, in addition to the coordinates of A and B. Is this information sufficient to draw the arc (i.e. find its center point)?
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Find the line that is closest to 4 skew lines

If I have 4 skew lines in $\mathbb{R}^3$, how can I find the line $L_c$, that is closest to all of them? I know that with 3 skew lines, there is always a line that intersects all of them, in fact ...
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2answers
45 views

Does the quadrilateral have an inscribed circle?

Question: Let o1, o2 be circles inside an angle tangent to one of its sides in points A, B and to the other in points C, D. Prove that if o1, o2 are externally tangent, then ABCD has an inscribed ...
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1answer
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Compute the angle in the quadrilateral

Question: Let $ABCD$ be a convex quadrilateral with $∠DAC = ∠ACD = 17^{\circ}$; $\angle CAB = 30^{\circ}$; and $\angle BCA = 43^{\circ}$. Compute $\angle ABD$. What I have so far: Since $\angle DAC ...
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0answers
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Find an angle without using trignometry

This question has already been asked before, so it is actually a duplicate of: How to make correct system of equations to solve for the angles in this triangle? But I was trying to solve this ...
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2answers
61 views

How does this make sense?

Just today I encountered a problem at brilliant.org: As we saw in the last problem, when multiplying sums and differences, each term in each sum or difference must be multiplied by every term in ...
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1answer
35 views

Prove all points lie on a common circle

Question: Let P, Q, R be points on the sides AB, BC, CA, respectively, of a triangle ABC. Assume that the circumscribed circles of the triangles PBQ and QCR intersect at points Q and S. Prove that the ...
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1answer
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Where does a chord of an Ellipse equal to the length of the minor axis but running parallel to the major axis cross the minor axis.

Given an Ellipse, I need to know where a chord equal to the length of the minor axis but running parallel to the major axis cross the minor axis.
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3answers
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Center of Area of an Annular Sector/Radius of Revolution

To generate a solid ring torus around the cylinder, the circle (2) is revolved around the cylinder along a path $2\pi R$, where $R = r_{cylinder}+r_{circle}$. To generate a solid rectangular toroid, ...
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1answer
38 views
+50

At least an ortogonal projection inside a side

Let $P$ a point inside a convex n-agon and let $P_1, P_2, ..., P_n$ the ortogonal projections of $P$ on the sides of the n-agon. How can I show that at least one of these projections lies inside a ...
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1answer
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How to find the direction cosines of the line which bisects the angle positive degrees between positive direction of Y and Z axes

I have a text book solution like - The line bisects angle between Y and Z axes. Therefore the line lies in the YZ plane. Hence the X-axis is perpendicular to the line. Now how come X-axis is ...
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16 views

Check for Point Collinearity and rearrange line

imagine you are having a 2D object, which looks like this: $$ l = (p_1,p_2,...,p_n) $$ with $p_i = (x_i, y_i)$. It may be possible that $p_1 = p_n$, or that $l$ only contains two points, which are not ...
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1answer
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Solving for Ellipse Parameters Given a radius and angle (Challenge 2)

Given an ellipse centered on the origin in an x-y plane expressed as $$\bigg(\frac{x}{a} \bigg)^2+\bigg(\frac{y}{b} \bigg)^2 = 1$$ In polar coordinates with radius $R$ and angle = $\theta$, this can ...
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1answer
46 views

The sphere with three ends?

The preceding image take form Matthias Weber's Classical Minimal Surfaces in Euclidean Space by Examples notes is called the sphere with three ends. But what does it have to do with a sphere and why ...
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1answer
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Solving for Ellipse Parameters given a radius and angle (Challenge 1)

Given an ellipse centered on the origin in an x-y plane expressed as $$\bigg(\frac{x}{a} \bigg)^2+\bigg(\frac{y}{b} \bigg)^2 = 1$$ In polar coordinates with radius $R$ and angle = $\theta$, this can ...
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1answer
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How to calculate rectangle tangent to sphere

Given a rectangle $ABCD,$ how do I calculate points $A, B, C, \; \text{and}\; D\;$ if I place the rectangle tangent to a sphere, centered at a given Latitude and Longitude, and given a "Rotation" ...
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2answers
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How to calculate area of an ellipse based on its formula?

How can I determine the area of a half-ellipse if all that is given is $y = \sqrt{1-n^2x^2}$? I have tried both geometry and calculus, but without convincing results… Thank you
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What is the difference between a “ line ” and a “ straight line ”?

Is there actually a difference between a line and a straight line ? Is figure 1 a line . ? . Should I take help from " Euclid "? I believe according to " Euclid " the above figure is a valid line.
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Finding axis of symmetry given n points and their surface normals

Given a roughly torus-shaped 3D object: it has cylindrical symmetry, one axis of rotation, and is symmetric with respect to any angle around that axis. Imagine a donut, but with any number of ridges ...
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In $\triangle ABC$, with $D$ on $BC$, if perpendiculars from $D$ to the other sides are equal, can we conclude that $\triangle ABC$ is isosceles?

In the figure, $\triangle ABC$ is a triangle. There exists a point $D$ on $BC$ such that when two perpendicular lines are drawn from $D$ to $AB$ and $AC$, respectively, $DE = DF$, where $E$ and $F$ ...
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Relation between angles made by position vector with coordinate axes [on hold]

Let $l, m$ and $n$ be the direction cosines of a vector We know that $l^2+m^2+n^2=1$ Let $\alpha, \beta$ and $\gamma$ be the angles made by position vector with coordinate axis. But what is the ...
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2answers
243 views

Homotheties: Let $A$ and $B$ be distinct points of a circle $o$. What is the set of possible centroids of triangles $ABC$ with $C\in o$?

Question: Let $A$ and $B$ be distinct points of a circle $o$. What is the set of possible centroids of triangles $ABC$ with $C\in o$? Here is what I have: The angle at $C$ will always be the same as ...
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0answers
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Geometry_doubt_anyone help [on hold]

Had a doubt in geometry, please anyone make it clear with proper explanation Thank you 🙂🙂 Please check this link for question https://drive.google.com/file/d/1I6yY3_ZR6DcIds7SL6CLtRiE_IC_JyrT/...
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1answer
39 views

faster than O(n^4) algorithm for solving the following problem

This question is quite similar to the previous one I asked here. I recommend taking a look at it and the solution given by @platty before continuing. Given a set of n inequalities each of the form ax+...
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1answer
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Relative homology and limit

Let $X$ be a smooth manifold and $O\subset X$ be an closed set containing a non-trivial neighbourhood of $x\in X$. The reason to ask the question is to clarify the relationship between limit and ...
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The formula for finding the number of sides in a hypercube. [on hold]

I am interested in finding the number of faces on an n-dimensional hypercube? Does such a formula exist, if so, where can I find it?
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Prove that a harmonic homology preserves the conic

I came across a question in the book by Judith N. Cederberg and I’m learning about projective geometry. One of the question was “Show that a harmonic homology whose centre and axis are pole and ...
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1answer
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Product of diagonals of a parallelogram

I came across a property which I am not sure if true in general. Suppose you have a parallelogram whose vertices are $v_1, v_2, v_3, v_4\in\mathbb R^2$. Let's say that the side $[v_1,v_4]$ is parallel ...
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How to determine what kind of conic section in the affine plane?

So, I've been struggling a bit with understanding this problem. Let $P^2$ be the real projective plane with homogenous coordinates $(x_0:x_1:x_2)$ Let $\cal{C}$ be the line given by $$x_0^2 + 2x_0x_1+...
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My calculus method of finding the circumference of a circle gives $16r-8r\sqrt{2}$. Where's my mistake?

To find the circumference of a circle using calculus, I've seen the approach of starting with $(ds)^2 = (dx)^2 + (dy)^2$, but my approach should work, too. I take a small slice of the circle $dθ$. I ...
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4answers
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Review on my method for $Number$ $of$ $diagonals$ in a regular $n$-gon is $\frac12n(n-3)$

I have an assignment on permutations and combinations topics. In that there is a question- The number of interior angles of a regular polygon is $150^\circ$ each. The number of diagonals of the ...