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
12 views

Find the length as a function of $r_1,r_2$

We are given two mutually tangent circles in the plane, with radii $r_1,r_2$. A line intersects these circles in four points, determining three segments of equal length. Find this length as a ...
1
vote
0answers
8 views

Intersection of a surface and a line.

I have a surface ($H$) that passes every corner points(one coordinate gets its maximum $1$ while others $0$), such as $(1,0,...,0), (0,1,0,...,0),....,(0,...,1).$ H is characterized by ...
1
vote
1answer
13 views

Signed angle difference without conditions

I've got two angles in $0 \leqslant a < 360$ and I need to find the signed difference between them which should be $-180 < \Delta < 180$. Is there a way to calculate the difference with ...
1
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0answers
32 views

Ratios of the major axes of ellipses inscribed in a triangle

I recently came across a question that went something like this: In a triangle with vertices (0,0), (6,0), (2,4) an ellipse is inscribed such that it has the largest area. Now it is dilated, that is ...
3
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3answers
56 views

What's the equation for a rectircle? (Perfect rounded-corner rectangle without stretching on only one dim)

The equation for a rounded square seems to be: $x^4 + y^4 = 1$ You can make the radii smaller by increasing (over the even integers) the exponents in the equation. Here's a picture: Wolfram Alpha ...
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0answers
19 views

Synthetic geometry , angles

Let $ABC$ be a triangle such that $m(\sphericalangle ACB)>30$ and $M$ in the interior of the triangle with $m(\sphericalangle BMA)=120, m(\sphericalangle BCM)=30$. Let$\{ D\} = AM\cap BC$ and $P ...
0
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1answer
28 views

Is the Locus circle?

Locus of points such that sum of it's distances of them from four fixed points remains constant? Is the locus circle? I was not able to solve it as there were four radicals. Is it a theorem?
1
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1answer
18 views

Distance between a point $X$ and a line $2x-y = 1$

I am asked to state a condition for a point $X$ such that the point $X$ is a distance of 10 units from the line $2x-y = 1$, and then find the Cartesian equation of the set of all points $X$ that are a ...
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4answers
61 views

How Many Times The Clock Hands Make an angle Theta?

You might have encountered such a question where $\theta=90^o$ but this a little different and is causing a little problem. I approached this problem in the following matter and solved for $\theta= ...
1
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2answers
29 views

Area of piece of paper folded around straight line of orientation $\theta$

Imagine drawing a straight line $l$ through the center of a square piece of paper with area $1$. Now fold the paper along that line. Q: What is the function for the area covered by the folded ...
2
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3answers
28 views

How to find position of point that is x unit distant from AB line segment and y unit distant from BC line segment?

I am trying to calculate coordinates of point P, which is x units distant from AB line segment and y units distant from BC line segment. Edit: I am trying to write code for general solution. As ...
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2answers
25 views

Question on geometry related to Trapezium and Isosceles Triangle

In figure $AD\perp DE$ and $BE\perp ED$.$C$ is mid point of $AB$.How to prove that $$CD=CE$$
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1answer
33 views

How to find another solution for two cone intersection?

Lets say we have two cones and they intersect: $$ \sqrt{(x-a_1)^2 + (y-a_2)^2} + a_3 = \sqrt{(x-b_1)^2+(y-b_2)^2} + b_3 $$ And we know one solution $x_1, y_1$ for this intersection. Is there any ...
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0answers
11 views

Property of all k-dimesional shapes

A close friend showed me an intuitive pattern that given a k-dimensional convex polytope P, if one doubles every sidelength to generate a $P'$ it is possible to fit $2^k$ copies of $P$ within the ...
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2answers
23 views

find the lenght of segment in a skew quatrilateral

Let $ABCD$ be rectangle so that $|AB| = x$ and $|BC| = y$. Suppose we fold the rectangle along the diagonal $BD$ so that the planes $ABD$ and $BCD$ will be perpendicular. What is the lenght of $|AC|$ ...
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0answers
20 views

given skew lines $l$ and $m$ find the geometric locus

Suppose we have two skew lines $l$ and $m$. I want to find the geometric locus of points $P$ for which there is not line passing trough $P$ intersecting $l$ and $m$. I know the locus of points should ...
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1answer
23 views

Let $O$ be the center of the circle circumscribed around a triangle $ABC$. Does $C$ belong to the plane where the points $A$, $B$ and $O$ belong? [on hold]

Let $O$ be the center of the circle circumscribed around a triangle $ABC$. Does $C$ belong to the plane where the points $A$, $B$ and $O$ belong?
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3answers
275 views

Why is the surface area of a sphere not given by this formula?

If we consider the equation of a circle: $$x^2+y^2=R^2$$ Then I propose that the volume of half of a sphere of radius $R$ is given by the summation of the circumferences of the circles between the ...
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0answers
21 views

Concurrence of lines through triangle midpoints [duplicate]

Suppose that points $A, B, C$ form a triangle and that $A'$ $\in$ $l_{BC}$ and $B'$ $\in$ $l_{AC}$ and $C'$ $\in$ $l_{AB}$ are such that the lines $l_{AA'}$, $l_{BB'}$, $l_{CC'}$ are concurrent at G. ...
1
vote
1answer
26 views

concentration of volume of hypersphere

I am reading about features of volume of hyperballs, where I see two theorems, Most of the volume of the d-dimensional ball of radius r is contained in an annulus of width $O(r/d)$ near the ...
1
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0answers
33 views

Concurrence through a triangle

Suppose that points A, B, C form a triangle and that A' $\in$ $l_{BC}$ and B' $\in$ $l_{AC}$ and C' $\in$ $l_{AB}$ are such that the lines $l_{AA'}$, $l_{BB'}$, $l_{CC'}$ are concurrent at G. ...
1
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1answer
19 views

Calculate most parallel vector

Let's suppose I have a vector $\vec{a}$ and arbitrary many vectors (in my example two: $\vec{b}$ and $\vec{c}$), which all have a common starting point P. Now I want to find out which of these vectors ...
2
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1answer
66 views

Would you confirm this as a proof to the Pythagorean theorem?

I'm new in mathematics, and trying to build my way up starting by doing simple tasks. My current one is proving the Pythagorean theorem without looking it up. This photo contains my current "proof" ...
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2answers
50 views

Areas of triangles in hexagon

A hexagon $ABCDEF$ with parallel opposite sides is given. Prove that $[ACE]=[BDF].$ (here $[]$ denotes area of triangle) Since the sides are parallel does that mean that it is equiangular as ...
12
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7answers
1k views

Length of a Chord of a circle

I was wondering about the possible values that the length of a chord of a circle can take. The Length of a chord is always greater than or equal to 0 and smaller ...
0
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2answers
27 views

Geometry-||gm proof

$ABCD$ is a parallelogram in which $P$ and $Q$ are the mid points of the sides $AD$ and $BC$ respectively. If $BP$ & $QD$ intersect the diagonal $AC$ at $X$ and $Y$ respectively then prove that ...
-4
votes
1answer
24 views

Find both the diagonals with area and side given [on hold]

Side of rhombus $= 65$ cm Area of rhombus $= 1024$ cm$^2$ Find the diagonals of the rhombus.
0
votes
1answer
31 views

Rotation addition with quaternions

My task is: "Describe rotation $S \circ R$ by axis and angle, where $R$ is rotation around $(0,1,1)$ by 90 degrees, and $S$ is rotation around $(1,-1,0)$ by 90 degrees." I should use quaternion ...
0
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0answers
25 views

Prove that $x+y \tan {\frac{(\alpha + \beta)}{2}}=0$ [on hold]

The line joining $A(a\cos{\alpha},b\sin{\alpha})$ and $B(a\cos{\beta},b\sin{\beta})$ is produced to the point $M(x,y)$ such that $AM: BM=b:a$; prove that $x+y \tan {\frac{(\alpha + \beta)}{2}}=0$ ...
0
votes
1answer
34 views

If the coordinates of the vertices $B$ and $D$ are $(7,3)$ and $(2,6)$ respectively, find the coordinates of the vertices $A$ and $C$.

The sides of the rectangle $ABCD$ are parallel to the co-ordinate axes. If the coordinates of the vertices $B$ and $D$ are $(7,3)$ and $(2,6)$ respectively, find the coordinates of the vertices $A$ ...
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0answers
13 views

How to visualize the product of two segments lengths? [duplicate]

So there seems to be easy ways to visualize addition. If three points a, b, and c are on the same straight line respectively, we can say that the sum of the lengths of ...
2
votes
5answers
66 views

How limiting/ heavy is the “triangle inequality” assumption?

Suppose a theorem proves something about a family of distance measures, with this the triangle inequality assumption. How limiting this assumption is in reality? What are some real-world examples of ...
0
votes
0answers
12 views

Translate and Rotate mesh

I have a mesh constituted of some vertices in 3d space, let's call them $(x_1,y_1,z_1),(x_2,y_2,z_2),\cdots,(x_n,y_n,z_n)$. The mesh's central point is $(0,0,0)$. How to find out the new coordinates ...
0
votes
1answer
21 views

The Name of a Polyhedron with 6 Quadrilateral Faces, 8 Vertices, and 12 Edges

(Don't say 'cube' or 'rectangular prism') I'm looking for a generic name for polyhedra with 6 Faces, 8 Vertices, and 12 Edges where each face could be any quadrilateral shape: rectangle, rhombus, ...
2
votes
1answer
24 views

Continous family of $n$-gons

The statement of this problem asks to show that if $A$ and $B$ are two distinct convex $n$-gons there is a continous family of convex $n$-gons such that the first in that family is $A$ and the last is ...
2
votes
1answer
15 views

Two new conics associated with two triangles inscribed a conic

1-Let $ABC$ and $A'B'C'$ be two triangle inscribed a conic. Let $P$ be a point on the conic. $PA$ meets $B'C'$ at $A_1$, define $B_1, C_1$ cyclically. $PA'$ meets $BC$ at $A'_1$ define $B'_1, C'_1$ ...
2
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0answers
21 views

Bound on “width” of points in a plane

Suppose we define width $w(P)$ of point set $P$ in a plane to be the ratio of the maximum distance to the minimum distance between the points in $P$. (Assume unique coordinates so that $w(p)$ is ...
0
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0answers
19 views

How to prove that a convex polygon is cyclic

Is there an easy way to tell if a convex polygon is cyclic? I was told that if the vertices of the $n$-gon are $A_1,A_2,\ldots,A_n$, it is enough to prove that $A_1A_2A_3A_i$ is cyclic for each ...
0
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0answers
18 views

Using oblique projection can you always rotate a triangle to look like an equilateral triangle? [duplicate]

Starting with any triangle using oblique projection, can you view any shape triangle from an angle to see it as an equilateral triangle?
2
votes
2answers
26 views

Regular/Right hyperbolas through three points

Find all equations of regular hyperbolas passing through the points $A(\alpha,0),B(\beta,0),$ and $C(0,\gamma)$. Attempt: I am assuming here that regular means right. We can then write the ...
0
votes
1answer
11 views

Having trouble drawing an arc using atan2 and specified range of arc.

I'm writing code. My arc invariant is: $$ \theta_0, \theta_1 \in [-2\pi, 2\pi], \\ \theta_1 - \theta_0 \leq 2\pi $$ where $\theta_0$ is the arc beginning angle and $\theta_1$ is the arc end angle. ...
0
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1answer
15 views

scalene trapezoid point of diagonals intersection

We have a scalene trapezoid. We know AB and CD bases and the diagonal AC. Be P the point of intersection of the two diagonals. Is it possible to find the general expression for AP?
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0answers
16 views

How to rotate a 3D object, using only local x-, y-, and z-rotations, so that it always faces a camera at the origin

I have been struggling with a difficult problem involving 3D rotations. I first came across this problem in a computer science context, but I've attempted to generalize it a bit before posting. (I ...
5
votes
1answer
48 views

$\pi$ is dependent on properties of geometry, assuming that we define it as $C/d$. Could then the $\pi$ also be an integer?

$\pi$ is dependent on properties of geometry, assuming that we define it as $C/d$. Could there be a geometry where $\pi$ is a rational number or an integer?
2
votes
1answer
38 views

Lebesgue measure and similarities

There is a well-known theorem in Euclidean geometry (Eucl. VI-19) that says that the ratio of the areas of any two similar polygons is equal to the square of the corresponding ratio of similarity. ...
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0answers
31 views

How do I group points into several circles with a given radius?

There is a set of points on the Cartesian coordinate system. How do I group points into several circles with a given radius and meeting the following requirements: each point belongs to a group at ...
0
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0answers
21 views

Line postulate in Geometry

It is a postulate that it is possible to draw a straight line from any point to any point. Firstly, if the points are the same does this postulate still hold? Secondly, if this is a fact that can't be ...
0
votes
2answers
41 views

Question on Geometry and cyclic circles

Two circles intersect at $P$ and $Q$.Through $P$ two lines $APB$ and $CPD$ are drawn to intersect circles at $A,B,C,D$. $AC$ and $DB$ when produced meet at $O$. How do I prove that $OAQB$ is cyclic ...
0
votes
1answer
37 views

Find the height $h$ of a circular segment based on the Radius $R$ and length $c$

I have found the formula for calculating the R of a circle, based on a circular segment, which is: $$ R=\frac{h}2+\frac{c^2} {8h}$$ where $R$ is radius, $c$ is the length of the segment, and $h$ is ...
0
votes
1answer
33 views

disk-disk intersection area

I have two disks of radii $R_1, R_2$ with distance between centers, $d < R_1 + R_2$. How can I find the surface area common to the two disks? Rationale: Solar irradiation / energy input in ...