For questions about geometric shapes, congruences, similarities, transformations, as well as the properties of classes of figures, points, lines, angles.

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Pentagon Inscribed Circle Radius

Square ZENG has a perimeter of 12, with midpoints Y and I on sides ZE and ZG, respectively. Find the radius of the largest circle that can be inscribed in pentagon GIYEN. I got past all the simple ...
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1answer
9 views

Cyclic Hexagon Circumradius

A cyclic hexagon has side lengths of 2, 2, 7, 7, 11, 11, in that order. Find the length of its circumradius. Not sure if there is a theorem or formula for this, but I tried dividing it into 30°, 60°, ...
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14 views

Rectangular Prism

A rectangle with dimensions 6×8 is revolved about its longer side. Find the volume of the resulting finger. I've tried "revolving" it to be a cylinder and triangular prism, neither of which got me ...
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1answer
10 views

Volume of a Cone made from circle

Circle X has a radius of 15. Points D and F are located on circumference of the circle. Given that angle DXF measures 48°, find the volume of the cone that is formed by aligning the two straight ...
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19 views

prove in any traingle ABC if A>B then a>b

A,B are angles and a,b are sides How do you prove this using the sine rule? if A>B do you consider separate cases when A is acute and obtuse? also how do you prove the converse if a>b then A>B
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14 views

Height of a Trapezoid given diagonals

The diagonals of a trapezoid have lengths 17 and 15, and the segment connecting the midpoints of the bases has a length of 4. The height of the trapezoid can be expressed as (x√y)/y Find x + y. I ...
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1answer
18 views

$−8\sin 3x+5\cos 3x=4.3$ for $0<x<360.$

can you tell me please how to solve $−8\sin3x + 5\cos3x = 4.3$ for $0<x<360$? I find 6 solutions! and I don't know if they are the correct, although they seem to fit in the equation! thanks!
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3answers
19 views

Finding the set of values of y for which there are no points on a curve

A curve has the equation: $$y=2x-5+ \frac{18}{x+4}$$ Find the set of values of $y$ for which there are no point on the curve. What I did: Found the maximum and minimum points: ...
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2answers
10 views

Find the equation of the locus of the mid-point between an elliptical point and its directrix

I'm struggling with this question: The point $P$ lies on the ellipse $x^2+4y^2=1$ and $N$ is the foot of the perpendicular from $P$ to the line $x=2$. Find the equation of the locus of the ...
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1answer
35 views

Inequality of area of two triangles

Let $ABC$ be a triangle with sides $a,b,c$ and $A_1B_1C_1$ be another triangle with sides $a+\frac{b}2$, $b+\frac{c}2$, $c+\frac{a}2$. Prove that: $$\frac94[ABC]\le[A_1B_1C_1]$$ I tried using ...
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8 views

Pappus Configuration with parallel lines

Let $A,B,C,D,E,F$ be points such that $A,C,E$ are collinear and $B,D,F$ are collinear with $AC\parallel BD$, also $AF\parallel CD$. Let $L = AB\cap DE$ and $M = BC\cap EF$. Prove that $LM \parallel ...
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9 views

Find the x-coordinate as the chord of two points on a parabola touches the x-axis

The chord joining the points $P(ap^2,2ap)$ and $Q(aq^2,2aq)$ on the parabola $y^2=4ax$ has the equation $(p+q)y = 2x + 2apq$. A variable chord $PQ$ of the parabola is such that the lines $OP$ and ...
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0answers
19 views

Series and sequences significance [on hold]

What is the significance of series and sequences like trigonometry series in nature
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3answers
50 views

Maximum value of $\sin A+\sin B+\sin C$?

What is the maximum value of $\sin A+\sin B+\sin C$ in a triangle $ABC$. My book says its $3\sqrt3/2$ but I have no idea how to prove it. Can anyone help? :)
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1answer
9 views

Finding the equations of the tangents to a circle with center $(3a,0)$

I have this question (which I've redacted): A point P moves in the x-y plane so that its distance from the origin, O, is twice its distance from the point with coordinates $(3a,0)$. If the ...
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1answer
16 views

How do I correct the position of my robot in this diagram? Having a hard time visualizing the formulas needed

Okay, so say that I have a robot which travels in a grid of tiles. Between each tile is a black line. At both sides of my robot is a light sensor. The robot only travels to adjacent tiles. Knowing ...
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0answers
28 views

Search Patterns. The Circling of The Circle and The Squaring of The Square.

Consider these two search patterns. {1} A square moving in straight lines in what you might call a "square-spiral" pattern, covering an infinitely large square. {2} A circle spiraling out covering ...
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28 views

riammanam manifold [on hold]

Lemma 4.6. Let ∇ be a linear connection on M . There is a unique con- nection in each tensor bundle T k l M , also denoted ∇ , such that the following conditions are satisfied. ( a ) On TM , ∇ agrees ...
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1answer
21 views

Calculate X Y Z from two specific degrees on a sphere

I am a programmer, don't know much about advanced math. I would need the exact formula(s) that could achieve this, so I can translate it to my programming language. I am having a headache trying to ...
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1answer
27 views

Circle problem and area [on hold]

What is the area of the circle centered at the origin with radius $5$, restricted to the domain where $x>0$ and $y>0$.
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1answer
21 views

Are congruent shapes similar, too?

I know that congruent shapes have the same size and the rotations don't matter. I just want to know if congruent shapes can also be similar at the same time. I've seen that similar shapes have one ...
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0answers
23 views

Compare time it takes to travel a curve and a line

Suppose you have a right triangle ABC with hypotenuse AB, AC is along gravity direction, C is the right angle. c1 is AB, c2 is a smooth and convex curve within the triangle connecting A and B. You ...
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0answers
25 views

How to check if a set of coordinates creates a polygon?

Well, hello. I've got a set of coordinates and i want to check if it creates one polygon or 2 (or more) polygons. Coordinates are being read from input stream, one after another, for example: ...
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3answers
37 views

A tangent circle element between 2 intersecting vectors

2 Vectors, which are originating from one point I. I want to the replace the sharp corner (I) with an arc (circle element) with a radius of r. The arc touches the vectors at T1 & T2. What is the ...
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0answers
10 views

verifying the geometric relationship in the Blinn-Phong reflection model.

I would like to prove to myself that the angle between the normal and the half vector is twice the angle between the reflection vector and the eye vector, as shown in this image taken from the ...
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0answers
14 views

Equal time path

Given two points A and B, where A is at a higher position than B. It's easy to find the time it takes a mass point to travel from A and B over a straight slope under gravity. Now can you find a smooth ...
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1answer
27 views

Time it takes a mass point to go down a curve under gravity

An age old question. How to calculate the time it takes a mass point to go down a frictionless curve under gravity? P.S. The curve is convex and smooth and can be of any kind of shape.
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2answers
40 views

Proving a triangle equilateral given condition $al_a^2+bl_b^2+cl_c^2=9R\Delta$

$ABC$ is a triangle, with $l_a$, $l_b$, $l_c$ as angle bisectors, $R$ as circumradius and $\Delta$ as area, such that: $$al_a^2+bl_b^2+cl_c^2=9R\Delta$$ Is it true that $ABC$ is equilateral? I am ...
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1answer
16 views

Two circles externally tangent find distance from the center of smaller circle to the point of tangency.

The radius of two circles that are tangent externally to each other is $r$ and $s$.Suppose $r>s$ and two outer tangent of circles intersect at point $P$.Denote the center of smaller circle by ...
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0answers
18 views

Affine Buildings

I am trying to study affine buildings. So far I learn a lot of theoretical properties and definitions, but it was hard for me to find a good example of this object to "visualize" the theory. (Yes, I ...
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1answer
38 views

To calculate side of the Equilateral triangle

The figure is an equilateral triangle. 3 line segments , which meet at a(any) point in the triangle , are of the length 5cm, 4cm, and 3 cm as shown in the figure. Find the side of the equilateral ...
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2answers
47 views

Quaternion - Spinor relationship?

I've known for some time about the rotation group action of the ('pure') quaternions on $ \mathbf{R}^3 $ by conjugation. I've recently encountered spinors and notice similarities in their definitions ...
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2answers
53 views

Optimal Box-in-a-Box-in-a-Boxing

As inspired by this closely related problem, suppose I have $n$ cuboid boxes, all with arbitrary (possibly random) finite dimensions. For any two boxes, $B_1$ with dimensions $w_1,h_1,d_1$, and $B_2$ ...
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0answers
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How is distance between two points defined in barycentric coordinates?

Hope someone can help. I have this 3-d simplex (a tetrahedron) and its vertexes have barycentric coordinates defined as follow: $A_1=(1,0,0,0), A_2=(0,1,0,0), A_3=(0,0,1,0), A_4=(0,0,0,1)$. I am ...
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1answer
17 views

Compute direction of a cylinder by using 10 coefficients

I am wondering if anyone knows how to compute the direction of a cylinder by using the 10 coefficients. For example, we have the equation of a cylinder as ...
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1answer
16 views

Create a Ray from two points

I know it's too easy for this website but I couldn't think of it myself. I have a point A(x1,y1,z1) and another point B(x2,y2,z2). And I represent a ray like this : r(t) = o + t *d. Using the given ...
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1answer
13 views

Gauss Curvature…Product of Minimum and maximum values

The function g(ϑ ) = cos2 (ϑ ) fxx (x0 , y0 ) + 2 cos(ϑ )sin(ϑ ) fxy (x0 , y0 ) + sin2 (ϑ ) fyy (x0 , y0 ) represents the Gauss curvature of the surface f (x, y) at the critical point (x0 , y0 ) in ...
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0answers
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Comparing/contrasting hyperbolic and Euclidean geometry - or, on how ${\rm PSO}_2(\Bbb R)$ sits inside ${\rm PSL}_2(\Bbb R)$

I am studying hyperbolic geometry, in particular comparing and contrasting it with familiar Euclidean geometry. Let $\Bbb E$ be the Euclidean plane, and $G={\rm Iso}^+(\Bbb E)$ be the group of ...
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1answer
13 views

Trapezoids and Bases [on hold]

A trapezoid has bases of length $x$ and $4$. Let $P$ and $H$ be points on opposite legs of the trapezoid. $PH$ is parallel to the bases and divides the trapezoid into two quadrilaterals of the same ...
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2answers
53 views

About the term $-\nabla_{[u,v]}w$ in the definition of Riemann curvature tensor

As we know, in the definition of Riemann curvature tensor, we require $$ R(u,v)w=\nabla_u\nabla_v w-\nabla_v\nabla_u w-\nabla_{[u,v]}w $$ Could somebody tell me why we need $-\nabla_{[u,v]}w$ ...
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1answer
44 views

Square is a parallelogram?

I remember, in the geometry class, our teacher used to tell us some definitions or something that i don't really know about. Why is square a parallelogram?
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1answer
17 views

intersection of an ellipsoid and cylindrical plane.

I need to understand if an ellipsoid and a cylindrical arc intersect, what will be the general equation of the cutted ellipse? How can I solve for that equation? I know in 3D, the equation of an ...
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5answers
235 views

How can I find the radius of a circle from a chord and a section of the radius?

Draw a circle with center O. Line AD is a chord that is 8cm long. The arc above is smaller than the one below. B is the center of AD. Line CB is a line that is 2cm long. It meets AD at 90°. ...
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Problem with an inclined cone and planes

From the image given below, I want to prove that there exists a unique plane $p \neq P$ s.t. $p \cap$ inclined cone $=$ circle centered at $O_{2}$. I also want to prove that if ray $SO_{1}$ (where ...
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2answers
20 views

Creating a parametric Equation when given the points of a collinear line?

$(-70, 3)$, $(88, 81)$, and $(246, 159)$ are three collinear points. Write parametric equations for $x$ and $y$. (In other words, write equations that produce points when $t$-values are assigned.) ...
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21 views

Find planes of a solid with all lines

I'm searching to obtain all planes of a solid out of its corresponding lines. The lines are composed of two connecting points and this is all the information that I've got. What is the best way to ...
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1answer
22 views

Prove that the locus of a point P is a circle

I'm struggling with this geometry question: The fixed points A and B have coordinates $(-3a,0)$ and $(a,0)$ respectively. Find the equation of the locus of a point P which moves in the coordinate ...
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1answer
13 views

Let $A,B,C,D$ be the vertices of a four sided polygon taken in anti clockwise. Given $|AB|=|BC|=3,|AD|=|CD|=4,|BD|=5$ , Find $|AC|$

Let $A,B,C,D$ be the vertices of a four sided polygon taken in anti clockwise. Given $$|AB|=|BC|=3,|AD|=|CD|=4,|BD|=5$$ Find $|AC|$ My try:I have noticed trangles $ABD$ and $BCD$ are right ...
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1answer
63 views

Parametrising the unit circle without sine and cosine

Is there a nice way to make a smooth and periodic parametrisation $\gamma\colon\mathbb R\to S^1$ of the unit circle $S^1$ in $\mathbb R^2$ that does not somehow involve sine/cosine or (what I find to ...
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0answers
38 views

Is there a surface S$\subset R^3$ whose Gaussian curvature is -1 at each point S?

Is there a surface $S\subset \Bbb R^3$ whose Gaussian curvature is $-1$ at each point $S$? At first I think this does not make a sense. But googling and googling.. I found a 'final exam problem' ...