Tagged Questions

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

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-3
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0answers
15 views

Help me with this question

Length of AB, BC and CD are equal. length of AD=9,AE=6. Find the length of $CE^2$
1
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2answers
16 views

Display cans of food in a square-based pyramid.

Full question: The manger of another grocery store asks a stock clerk to arrange a display of canned vegetables in a square-based pyramid (top is one can, 4 cans under then, 9 cans under top 2 ...
1
vote
0answers
27 views

I need help with this geometry question.

Let ABC be a triangle with AB=AC. If D is the midpoint of BC, E is the foot of the perpendicular drawn from D to AC and F the mid-point of DE, prove that AF is perpendicular to BE. (JEE-1989) I ...
1
vote
2answers
16 views

How to prove that two lines of a shape are parallel?

∠A=∠B, and P is a point on AB such that AD=PD. Prove that DP is parallel to CB. Please state how also ($SSS$, $ASA$, $SAS$, $CPCTC$, Isosceles Biconditional, interior angles, etc.)
1
vote
2answers
21 views

How to prove that two lines of a quadrilateral are parallel?

Given that AC bisects ∠DAB and that AB=BC, prove that AD is parallel to BC. I need to know why for each statement (e.g. SSS, ASA, SAS, CPCTC, Isoceles Biconditional)
1
vote
0answers
13 views

Calculate area defined by matrix equation

Suppose I have an $n$-dimensional matrix $A$. I define a region as being the set of all vectors ${\bf x}$ such that when I calculate $A.{\bf x}$ the resulting coordinates are all between 0 and 1. (or ...
0
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0answers
8 views

Stereographic Projection preserves angles at the south pole

Show that stereographic projection preserves angles at the "south pole" $S=(0,0,-1)$. I really don't know how to approach this problem. The main problem is that I do not have a good definition ...
0
votes
0answers
19 views

Find points in a reference unit square

First of all sorry if this question has been answered or is in the wrong place. As part of an algorithm, I need to map two points $(P_0,P_1)$ in an arbitrary quadrilateral to a reference unit square ...
0
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0answers
9 views

Triangle Section Side Lengths

Point D is on side BC of triangle ABC, with AB=3, AC=6, and angle CAD = angle DAB = 60 degrees. What is the length of AD?
2
votes
0answers
15 views

A condition for both cyclic and tangential quadrilaterals

I'm looking for a nice condition that characterizes quadrilaterals that are both cyclic and tangential (i.e. there exists a circle that touches each side). I know that both concepts have some nice ...
-1
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0answers
22 views

Logical Geometry Challenge [on hold]

d the asd numdsber of easdddges be thasdat such a asdadaigure casdll asdn
0
votes
2answers
24 views

Side Section Lengths in a Right Triangle

Right triangle ABC has its right angle at C. Let M and N be the midpoints of AC and BC, respectively, with AN=19 and BM=22. What is AB?
0
votes
1answer
23 views

Hexagon Mayhem: Geometry Challenge

Suppose that there is a regular hexagon $UVWXYZ$. Suppose that $A$ is the region that is common to the triangles that are formed $VYZ$ and $UWX$. If the area of $A$ is 19, find the area of the regular ...
-4
votes
2answers
29 views

TRIANGLE Altitude Problem [on hold]

The sides of a triangle have lengths 15, 20, and 25. What is the length of the shortest altitude?
1
vote
1answer
11 views

Linear programming - geometric change between canonical and standard forms

Suppose that we are given a LP in canonical form, that is in the form $\{x \in \mathbb{R}^d |\ Ax \geq b \}$ and that we want to convert it to an equivalent LP in standard form $\{x \in \mathbb{R}^k \ ...
0
votes
0answers
21 views

Central Angle of a triangle

Does all three central angles of a triangle being 120 degrees imply that is it equilateral? I am trying to use this to prove that the line segments connecting an arbitrary point inside a triangle ...
1
vote
0answers
28 views

Does this notion of the “directed area” of a closed curve in $\mathbb R^3$ have a standard name?

Given an oriented surface $\Omega$ in $\mathbb R^3$, consider the quantity $\mathbf A(\Omega)=\int_\Omega\hat n\,\mathrm dA$. We may call this the "directed area" of the surface because, when $\Omega$ ...
-1
votes
0answers
8 views

mobius tranformation on the extended complex plane

write the result of translating by 4+i and then carrying out the inversion as a single mobius transformation on the extended complex plane? Someone help please...
1
vote
0answers
16 views

fluid flow through an orfice [migrated]

Forgive me for my ignorance. What would be the method to determine the pressure a non compressible fluid creates when forced though an orifice? Keep in mind this orifice does not have a constant ...
0
votes
1answer
25 views

Area of a triangle inside a larger triangle

It's been a while since I've done any geometry so I'm a bit confused by this question. We have a triangle $\triangle PQR$ whose total area is $90 \mathrm{cm}^2$. Another triangle $\triangle PTU$ is ...
-3
votes
1answer
35 views

geometry problem [on hold]

Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.
0
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1answer
24 views

Find the equation of the ellipse with given foci and $a$

I have to find the equation of the elipse with foci: $$(-1,-1),(1,1)$$ and $a = 3$ I could do it using the definition of elipse, which I know how to work with. But I need to do it using translation ...
2
votes
2answers
29 views

Sum of squares of side lengths of a regular polygon [on hold]

Given $n$ vertices, $P_i$, $i \in \{1\dots n\}$ of a regular polygon on the unit circle (radius $R = 1$), calculate the sum $|P_1P_2|^2 + \dots + |P_1P_n|^2$.
0
votes
1answer
18 views

geometry proof - Generalities of Geometry

Can you give a hint on how to prove that AB + CD > AF + FC +DE + EB?
0
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0answers
8 views

path metrics without geodesics

This is a follow-up of this question. Recall that a metric space $(X,d)$ is called a path-metric space if the distance between any two points in $X$ equals the infimum of lengths of paths between ...
8
votes
1answer
82 views

A geometric assembly: Triangle, circle, square, pentagon.

Let say we have an equilateral triangle and I draw its circumscribed circle, to continue we draw a square in which the previous circle is inscribed. After that we draw the circle circumscribed to the ...
1
vote
1answer
32 views

Do closed submanifolds have nonzero curvature on a non-zero-measure set

If $M\subset \mathbb{R}^n$ is a $m$-dimensional submanifold, does the set with non-zero gaussian curvature always have $m$-dimensional Hausdorff measure greater than zero? For a manifold homeomorphic ...
4
votes
0answers
17 views

Bounded area for any triangle formed by polygons

Let $P_1,P_2,P_3$ be closed polygons on the plane. Suppose that for any points $A\in P_1$ (meaning $A$ can be inside or on the boudary of $P_1$), $B\in P_2,C\in P_3$, we have $[ABC]\leq 1$. Is it ...
0
votes
1answer
17 views

Finding Orthocenter in Coordinate Geometry

If a triangle is formed by the equations \begin{gather}2x+3y-1=0\\ ~~x+2y-1=0\\ ax+by-1=0\end{gather} and has its orthocentre at origin, then what are the values of $a$ and $b$? (Please also tell me ...
1
vote
0answers
33 views

What is the average distance from the base of a rectangular pyramid to its apex?

What is the formula for the average distance from the base of a rectangular pyramid to its apex? For example, if the base of the pyramid is 30 feet by 8 feet, and the height of the pyramid is 12 feet, ...
3
votes
0answers
16 views

Product to vertices in triangle maximal

Suppose we're given a triangle $ABC$. At which interior point $T$ is the product of distances $|AT|\cdot |BT|\cdot |CT|$ maximal? Is it a known point, like the centroid or incenter?
0
votes
0answers
15 views

Abother question on game theory [duplicate]

There is a polygon with $N$ vertices drawn in the plane. The polygon is strictly convex, i.e., each internal angle is strictly smaller than 180 degrees. The vertices of the polygon are numbered 1 ...
0
votes
1answer
21 views

Understanding why Euler's Formula applies to planar graphs

I'm trying to prove that given a planar graph (by that I mean a graph where every pair of points is joined without crossings) $V-E+F = 2$. I can prove this by induction directly on the edges except ...
4
votes
1answer
62 views

Selecting a Random Point Inside a Cube

A point $P$ is selected at random inside a cube. Find the probability that $\angle APB \geq 135^o$, where $\overline{AB}$ is a body diagonal of the cube. I am not able to come up with the right ...
2
votes
2answers
59 views

Closed-form of $\tan(\frac{\pi }{9})$.

How can I get the closed-form of the above expression.I tried to find it but I can't do.
1
vote
1answer
40 views

Finding circle with two points on it and a tangent from one of the points

Two points P1(x1,y1) and P2(x2,y2) are known. In addition, a line slope passing through P1 is known. The aim is to construct a circle (or circular arc) that it passes through both P1 and P2 and it is ...
0
votes
2answers
16 views

What is the angle of a space diagonal in a cuboid?

What is the angle ($\sigma$) from $D$ to $R$, expressed as function of $R$, $T$ and $H$?
0
votes
1answer
17 views

Homeomorphism from $[0,1]\times[0,1]$ to $\overline{D}(0,1)$?

I'm trying to construct a homeomorphism from $[0,1]\times[0,1]$ to $\overline{D}(0,1)$. I'm pretty sure there is one. I've been trying to work geometrically : mapping $[0,1]\times[0,1]$ to ...
2
votes
2answers
57 views

Methods for calculating $\pi$ that use the sphere?

The area of the unit circle is $\pi$ and its circumference is $2\pi$. Consequently, many elementary methods for calculating and approximating $\pi$ use a geometric approach on the circle, such as ...
1
vote
1answer
23 views

A clarification on a problem on proving existence of regular polygons

So I was looking at this thread: How can I prove the existence of an octagon/decagon/dodecagon?, and while this question intrigues me I also have no way to solve it, and it was unanswered in the ...
0
votes
2answers
27 views

How to calculate the curvature of a curve whose equation is not given.

I want to calculate the curvature of the following curve (in blue) whose equation is not known. I shall be thankful.
0
votes
3answers
35 views

How to show whether three points in $\mathbb{R^4}$ lie on a straight line?

If you are given the coordinates of three points in $\mathbb{R^4}$. (Call these three points: A, B and C). How do we know whether these three points lie on a straight line or not? One way I can think ...
0
votes
1answer
19 views

Spherical Sector Volume

I'm trying to find the volume of a spherical sector without knowing the height of the cap. Wikipedia provides this formula: And says: "where φ is half the cone angle, i.e., the angle between the ...
0
votes
1answer
26 views

Problem with shaded regions in a square

The sides of a square are 16 cm in length. The midpoints of the sides of this square are joined to form a new square and four triangles (diagram 1). The process is repeated twice as shown in the ...
0
votes
1answer
29 views

Prove: sum of angles in triangle is invariant

Prove that the sum of angles in any plane triangle is invariant, without invoking the fact that the sum of angles in a triangle is two right angles. I've tried. Is it even possible?
3
votes
3answers
58 views

Why are randomly drawn vectors nearly perpendicular in high dimensions

I am struggling understanding this finding. Can somebody explain intuitively why randomly drawn high-dimensional vectors will tend to be mutually orthogonal? I realize that intuition in high ...
4
votes
0answers
52 views

Finding an angle between side and a segment from specified point inside an equilateral triangle

Here is the question: $\overset{\Delta}{ABC}$ is an equilateral triangle. D is a point inside triangle. $m(\widehat{BAD})=12^\circ$ $m(\widehat{DBA})=6^\circ$ $m(\widehat{ACD})=x=?$ I managed to ...
0
votes
1answer
19 views

Prove that a convex $d$-polytope has at least $d+1$ facets

This seems trivial but I can't come up with a formal proof. I think there should be a way to do this inductively but I can't figure out how$\ldots$ Any help much appreciated
1
vote
2answers
39 views

Maximal points on an $n$-dimensional ellipsoid

For an $n$-dimensional ellipsoid in $\mathbb{R}^n$ centered at $\mathbf{0}_n$, defined by a set of vectors $\mathbf{v}_i$ giving the directions of the principle axes and $\lambda_i$ representing the ...
4
votes
1answer
49 views

prove this properties of triangles trigonometric question

The triangle $DEF$ circumscribes the three escribed circles of triangle $ABC$. Prove that $$\frac{EF}{a\,\cos A} = \frac{FD}{b\,\cos B} = \frac{DE}{c\,\cos C}$$