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

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9 views

Prove that $\frac{r_1}{r-r_1}+\frac{r_2}{r-r_2}+\frac{r_3}{r-r_3}=\frac{r_1r_2r_3}{(r-r_1)(r-r_2)(r-r_3)}$

Let $D,E,F$ be the feet of the perpendiculars from the incenter $I$ to the sides $BC,CA$ and $AB$ respectively. If $r,r_1,r_2$ and $r_3$ are the inradius of the triangle $ABC$ and radii of the circles ...
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0answers
15 views

Show that $a\sin 2\alpha+b\sin 2\beta+c\sin 2\gamma=0$

If the internal bisectors of the angles of the triangle ABC make angles $\alpha,\beta,\gamma$ with sides $a,b,c$ respectively then show that $a\sin 2\alpha+b\sin 2\beta+c\sin 2\gamma=0$ I tried to ...
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1answer
14 views

Given the end-vertices of two line segments, how do you calculate the point at which they intersect?

Given only the vertices of each line segment, and it's assumed they intersect, how do I calculate the point at which they intersect (in two and additionally three dimensions)?
2
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1answer
23 views

Euclidean Geometry (Potential Menelaus Theorem)

I have a strong suspicion that this problem applies Menelaus's theorem, but I can't see it. I also tried algebraic manipulation (such as trying to re-write BD/DC in terms of AB or CP), but to no ...
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2answers
40 views

proof of isosceles triangle?

How do you prove this isosceles triangle? Given line AC is congruent to line BC Prove: Angle A=Angle B I've gotten to the angle bisector and SAS(side- angle- side), and I believe there is one more ...
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2answers
24 views

Smallest convex polyhedron containing integer points of a cylinder

A cylinder has height $6$ and radius $3$. The centers of the two bases are $(0,0,0)$ and $(0,0,6)$. Find the volume of the smallest convex polyhedron that encloses every lattice point inside the ...
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3answers
34 views

Show that $\csc^n\frac{A}{2}+\csc^n\frac{B}{2}+\csc^n\frac{C}{2}$ has the minimum value $3.2^n$

Show that in a $\Delta ABC$, $\sin\frac{A}{2}\leq\frac{a}{b+c}$ Hence or otherwise show that $\csc^n\frac{A}{2}+\csc^n\frac{B}{2}+\csc^n\frac{C}{2}$ has the minimum value $3.2^n$ for all $n\geq1$. ...
2
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2answers
32 views

If the circumcenter of the triangle $ABC$ is on the incircle of the triangle,then prove that $\cos A+\cos B+\cos C=\sqrt2$

If the circumcenter of the triangle $ABC$ is on the incircle of the triangle,then prove that $\cos A+\cos B+\cos C=\sqrt2$ How should i attempt this question?I thought over it hard but could not ...
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0answers
25 views

What is the name of a cilinder-like object based on an ellipse instead of a circle?

If I project a circle over the Z-axis, I'll get a cilinder. If I project a square over the Z-axis, I'll get a parallelepiped. If I project an ellipse over the Z-axis, I'll get a... whatsitsname? I ...
2
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0answers
31 views

Geometry/Trigonometry Determine angle in a Triangle [duplicate]

Triangle ABC is isosceles with BC as base, AB=AC and Angle A=20 degrees. Points D and E lie on sides AB and AC respectively, such that D lies between A and B, E lies between A and C, angle BCD=50 ...
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0answers
14 views

Jacobi determinant for high-dimensional sphere inversion

I need to find the Jacobi determinant for the unit sphere inversion in $\mathbb{R}^n$, i.e. the map given by $f(x) = \frac x {|x|^2}$ for $x\in \mathbb{R}^n$. The main problem is to figure out the ...
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1answer
14 views

determine cube orientation given one side in a perspective projection

Suppose that we are given an arbitrary quadrilateral T that does not have any parallel edges. I want to draw a cube in a three-point perspective projection such that T is one of its sides. The ...
3
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4answers
37 views

$R$ is the midpoint of $MN$ and the points where $AC$ intersects $MD$ and $ND$ are $P$ and $Q$, respectively. Show that $PR=QR$.

In square $ABCD$, $M$ and $N$ are points on $AB$ and $BC$, respectively such that $\angle MDN=45°$. $R$ is the midpoint of $MN$ and the points where $AC$ intersects $MD$ and $ND$ are $P$ and $Q$, ...
3
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0answers
48 views

Very special geometric shape (No name yet?)

I suppose this geometric shape is something very 'special'. I cannot clarify in short about being 'special', but I think this shape stands together with such special shapes like square and hexagon. ...
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0answers
28 views

Trignometry, bearings yacht race question [on hold]

In a yacht race each yacht has to sail around a set of 4 buoys, and then return to the start line in order to finish. We will assume that the buoys are just points and the start line is also a ...
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0answers
15 views

book suggestions on homogeneuos coordinate and projective geometry [on hold]

Will any one introduce some good books about projective geometry and homogeneuos coordinates to me?
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0answers
10 views

About the interior of a polytope

Let us consider a polytope in $\mathbb{R}^n$ (in this context it must NOT be bounded) $\mathcal{P} = \{ x: A \cdot x \leq c\}$ for some matrix $A$. Let $\mathcal{I} = \{ x: A \cdot x < c\}$ be a ...
-1
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0answers
22 views

Straight lines and its applicaton [on hold]

What are the number of lines passing through (1,1) and intersecting a segment of length 2 unit between the lines x+y=1 and x+y=3 ?
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3answers
38 views

Equilateral Triangle Property

If the vertices of a triangle have integral coordinates how to prove that the triangle cannot be equilateral ?
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1answer
20 views

How to describe the transformation that changes French flag to Russian flag?

http://www.wolframalpha.com/input/?t=crmtb01&f=ob&i=Russia%2C%20France%20flags I presume it can be described two group operators, but I'm not sure how to come up with the formal description. ...
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1answer
12 views

Find the cost of leveling the triangular plot.Find the area using Herons formula

A Triangular Plot has sides $600$m .$640$m and $700$m.How much would the leveling cost if $1m^2=50$Rs. Solve the Problem using Heron's Formula.
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1answer
32 views

Determining North-South Line Via Watch Method: Theory & Reason [on hold]

I recently read that if you're in the northern hemisphere and have an analog watch, then you can point the hour hand at the sun and know that a south line lies between (bisection) the hour hand and ...
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2answers
15 views

confused on to leave in centimeters or convert to cubic centimeters

The volume $V$ of the cylinder is $65\pi \mathrm{cm}^3$. The height of the cylinder is $5$ centimeters. Use the formula $V = Bh$ to find the area of the base of the cylinder.
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0answers
16 views

What are the loci of points z which satisfy the following relations?

a) |Z-Z1|=|Z-Z2| b) 0< Re(iZ)<1 c) |z|=ReZ+1 d) Im((Z-Z1)/(Z-Z2))=0 The professor did not explain loci in class and the text does not have any examples, so I am completely lost.
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1answer
26 views

Is there any practical use of $0^\circ angles$?

An angle is defined as two rays $\overrightarrow{YX}$, $\overrightarrow{YZ}$ sharing a midpoint $Y$; the angle formed is called $\angle XYZ$ or simply $\angle Y$. Then, the measure of $\angle Y$ is ...
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1answer
36 views

Proving the AIP theorem?#2

How do you prove the AIP theorem? (if a pair of alternate interior angles formed by a transversal intersecting two lines m and n are congruent then line m is parallel to line n) I already know you can ...
0
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1answer
24 views

The Length of an edge of the cube

One dimension of a cube is increased by 1 inch to form a rectangular block. Suppose that the volume of the new block is 150 cubic inches, find the length of an edge of the original cube.
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2answers
36 views

Deriving the formula of the Surface area of a sphere

My young son was asked to derive the surface area of a sphere using pure algebra. He could not get to the right formula but it seems that his reasoning is right. Please tell me what's wrong with his ...
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4answers
48 views

A question on finding for the dimensions of a rectangle [on hold]

The diagonals of a rectangle is 8 m longer than it‘s shorter side. If the area of the rectangle is 60 square meters, find its dimensions.
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0answers
21 views

staircase length in Whitney's flat norm and Jenny Harrison's natural norm

Can someone provide the complete calculation for the length of a staircase as it converges to a diagonal line in Euclidean space in a sequence in which the number of steps goes to infinity between two ...
0
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0answers
22 views

Is a compact set in the interior of a cone contained in the intersection of all slightly perturbed cones?

Suppose compact set $S \subseteq R^n$ is in the interior of $x_0+C$, where $C$ denotes a solid convex cone in $R^n$ with the vertex at $0$. I am trying to prove that $\exists r>0$ such that $$S ...
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3answers
21 views

Finding a coordinate over a right angle in a triangle where the other two coordinates are known

a B ------- C \ | \ | \ | c \ | b \ | \ | \| A Alright, this is a triangle I have, and ...
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2answers
22 views

General equation of line that goes through center of a circle and a point

Given an arbitrary point $P$, at $(x_{1}, y_{1})$, is there a general expression of a line that goes through a circle of radius $r$ centered at the origin? I know there are infinite number of such ...
1
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1answer
26 views

I have a problem in loci [on hold]

$ABCD$ is a square with side length $4$ cm. A variable point $P$ moves inside the square so that $PA\leq 4$ cm, $PC\leq PA$ and the area of $ABP$ is $\leq 6$ cm$^2$. Construct $ABCD$ accurately and ...
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1answer
29 views

A geometry question on finding the area of cyclic quadrilaterals

The circumcircle of a cyclic quadrilateral $ABCD$ has radius $2$. $AC, BD$ meet at $E$ such that $AE = EC$. If $AB^2 = 2\cdot AE^2$ and $BD^2 = 12$, what is the area of the quadrilateral?
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1answer
30 views

Validity of my proof by contradiction of converse of Pythagorean Theorem.

So in, $\triangle ABC$ it is given that $AB^2=AC^2+BC^2$. Let us assume that $\angle C\neq{90}^{\circ}$. And let us make a perpendicular $AD$ to $BC$. Now, by the Pythagorean Theorem, in $\triangle ...
2
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2answers
33 views

When do circles on the sides of a triangle intersect?

If you have a non degenerate triangle and two of the sides are chords of two respective circles, then under what conditions do the circles intersect at two distinct points? I'm having trouble ...
0
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0answers
32 views

Is math, in the end, only geometry [on hold]

When thinking about the Universe, or "reality", Isn't every part of mathematics a tool for expressing something geometrically further down the line? Yes, every part of math is related, but isn't ...
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1answer
20 views

Tool for the partition problem with planar rectangles

The classical "partition problem" asks how many ways one can write a given natural number as a sum of smaller numbers. One variant of this would be to ask if a positive real number can be expressed ...
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1answer
22 views

Set invariant under reflections is a ball?

Say that $A\subset \mathbb{R}^n$ is measurable and of positive, finite measure. I'm trying to see if the following is true. If $A$ is invariant under all orthogonal reflections across $(n-1)$ ...
2
votes
4answers
102 views

Geometric proof of $QM \ge AM$

Prove by geometric reasoning that: $$\sqrt{\frac{a^2 + b^2}{2}} \ge \frac{a + b}{2}$$ The proof should be different than one well known from Wikipedia: DISCLAIMER: I think I devised such proof ...
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1answer
22 views

Finding the distance from a mirror

My friend sent me this problem, and my efforts to solve it have thus far been frustrated. I need some insight! Joe is 6 feet tall, and standing in front of a mirror that is at eye level, and is 3 ...
2
votes
1answer
14 views

Lines and projective isomorphism

Let $\rho : P(\mathbb{R^3}) \to P(\mathbb{R^3})$ be an homography, this is, a projective isomorphism induced from the isomorphism of vector spaces. I'm trying to understand what information about ...
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1answer
48 views

Frechet mean of $k$ elements in the n-dimentional sphere.

The Frechet mean of $k$ elements $x_1, \ldots, x_k \in S = \{ x \in \mathbb{R}^n,\, \| x \| = 1 \}$ is defined as the $\text{arg}\underset{\|x\|=1}{\text{min}} \sum_{i=1}^n d^2(x_i,x)$. Where ...
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1answer
59 views

How to calculate distance from the International Space Station given coordinates?

How would one calculate how far away a point is (latitude/longitude) from the international space station given its latitude/longitude/altitude? The distance would be direct as if drawing a straight ...
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0answers
43 views

Proof of Simple Properties of Volume

Let $e_{1},\ldots,e_{n}$ be vectors in $\mathbb{R}^{n}$. Define a parallelepiped $P$ to be a translate of the set $$\left\{x\in\mathbb{R}^{n} : x=t^{1}e_{1}+\cdots+t^{n}e_{n}, 0\leq t^{i}\leq ...
1
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1answer
49 views

Getting the angle that is needed for covering a given distance on an ellipse's cirumference

In a small programming exercise I asked myself, I want to calculate various things about ellipses. The part I'm stuck with is the following: I want to calculate the angle that is needed cor covering a ...
0
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1answer
32 views

Show that a point lies on the diagonal of quadrilateral

In a quadrilateral ABCD we choose a point E on the side AD and a point F on the side CD. Then we choose a point G on the line EF. Let H be the second point of the intersection between the circles that ...
1
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1answer
21 views

geometrical problem on triangle [on hold]

Given a triangle ABC, if $$a = \dfrac{2(b^2 - c^2)}{-b + \sqrt{b^2 + 4c^2}}$$, prove that $3m(C) = 2m(B)$.
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1answer
20 views

Write $CX,AY,BZ$ in terms of $CA,CB$ and the ratios $\alpha, \beta, \gamma$?

The point $X$ divides $AB$ in the ratio $\alpha$, $Y$ divides $BC$ in the ratio $\beta$ and $Z$ divides $CA$ in the ratio $\gamma$. Write $CX,AY,BZ$ in terms of $CA,CB,\alpha, \beta, \gamma$. I ...