1
vote
1answer
24 views

need explanation of what exactly is a directrix & focus?

((I'm not asking why do we need to know conic sections etc.) Like other similar questions.) I actually love math & currently learning conic sections in class, neither my textbook or teacher ...
2
votes
2answers
38 views

Orthogonal tangents to an ellipse [duplicate]

This is the problem I found back in the first year in the university. Suppose we have a non-degenerate (i.e. not a point and not an empty set) ellipse $E\subset \Bbb R^2$. Now define a set $D$ by a ...
1
vote
4answers
47 views

Equal perimeter and area

Find all triangles of which perimeter and area are numerically equal. I have got solution for right angle triangles but not of others
0
votes
1answer
48 views

How find this this distance $d_{1}d_{2}=b^2$

On the plane we have two points $A(\sqrt{a^2-b^2},0),B(-\sqrt{a^2-b^2},0)$ with $a>b>0$ and the line $L$, of which the equation is given ...
1
vote
1answer
25 views

Is there another way to solve the value field of a parameter of an line.

Assume $P$ is a point in line $x+y=m$, where $m \in \Bbb{R}$. There are two points $A,B$ in circle $$x^2+y^2 = 10$$ such that $PA$ and $PB$ are tangent lines of the above circle. If line: $x+y=m$ has ...
0
votes
0answers
17 views

Finding the equation of a plane by using point-to-point distances

Assume that we have a plane $P_1$ whose equation is known. I need to find the equation of plane $P_2$. If we choose a point set $N = \{n_1, n_2, ...\}$ on $P_1$ and another point set $M = \{m_1, m_2, ...
0
votes
1answer
38 views

The relation between the radiuses…

Find $\frac{R}{r}$ where $R$ is the radius of the circumscribed circle of a trapezoid and $r$ is the radius of the inscribed circle of this trapezoid. Thank you!
1
vote
1answer
55 views

Problem concerning inscribed and circumscribed circles…

Can you please help me solve this really difficult problem: Find R/r where R is the radius of the circumscribed circle of a trapezoid and r is the radius of the inscribed circle of this trapezoid. ...
0
votes
2answers
27 views

sides of two triangles which have different areas

consider 2 triangles like $\bigtriangleup ABC \quad and \quad \bigtriangleup \acute{A}\acute{B}\acute{C}$, which $S_{\bigtriangleup \acute{A}\acute{B}\acute{C}} \leq S_{\bigtriangleup ABC}$.(S stands ...
0
votes
2answers
52 views

How to show that a regular pentagon can't have all coordinates rational

This is pretty straightforward if we're allowed to use trigonometry, so I guess my question is Are there any nice (trigonometry-less) proofs of the fact that a regular pentagon in the plane must ...
0
votes
2answers
27 views

Find perpendicular vectors in subspace of $V_{3}$

Find all vectors of $V_{3}$ which are perpendicular to the vector $(7,0,-7)$ and belong to the subspace $L((0,-1,4), (6,-3,0)$. As a note, this is an extra question of a long exercise, the ...
0
votes
2answers
37 views

Prove $\sin^2 A = \sin^2 B \sin^2 C - 2\sin B \sin C \cos A$

I am asking for help with this proof: Given $\triangle ABC$. Prove that $\sin^2 A = \sin^2 B+ \sin^2 C - 2\sin B \sin C \cos A$
6
votes
0answers
63 views

Knight's metric: ellipse and parabola.

Knight's metric is a metric on $\mathbb{Z}^2$ as the minimum number of moves a chess knight would take to travel from $x$ to $y\in\mathbb{Z}^2$. What does a parabola (or an ellipse) became with this ...
2
votes
2answers
65 views

check if point is on a plane (using Heron formula ?)

Is this true that if any of parameters a, b, c, d is equal to sum of three others then 4 points are on same plane? I am given 4 points in 3 dimensional space. Is this correct to state that all 4 ...
1
vote
1answer
50 views

How to find out if four points are on the same plane, only by using distances?

There is a method called Cayley-Menger determinant in order to find if 3 points are collinear, 4 points are coplanar etc. provided that all the pairwise distances are given. However, in 2-D, there is ...
0
votes
0answers
23 views

Rotation of axes transformation as definition of vectors

Given a three-axes coordinate system ${1,2,3} $ by the right-hand rule, and a new coordinate system ${1',2',3'}$ , I know that one can define a vector $\vec{x}$ to be something that obeys the ...
1
vote
1answer
29 views

How to Create a Plane Inside A Cube

I have a $e \times e \times e$ cube and I want to create random planes with equation $ax + by + cz + d = 0$ inside this cube. I will put random points on those randomly created planes as well. Here ...
1
vote
1answer
10 views

Finding the degree to cover a flare shape

I have a lamp shade frame that I want to cover with bamboo slats. The top is 12" around and the bottom is 33" around. The distance from the top to the bottom most part is 6". What degree, with minor ...
2
votes
1answer
43 views

Interesting Analytic Geometry Problem: Find internal angles given coordinates of final point and length of line segments

I have been mulling over a really interesting question in analytic geometry that is much harder than it first appears to be. Hope you can provide some insight into solving it. If you only know: ...
0
votes
1answer
69 views

Differential geometry question.

Please explain how to solve this question. Thank you:) And sorry for hand-writing.
-1
votes
1answer
49 views

Rotate the Points on a Plane $P = ax+by+cz + d = 0$ parallel to $z = 0$ plane

I have a plane $P = ax+by+cz + d = 0$ and many points on that plane. I want to rotate $P$ so that it becomes parallel to $z = 0$ plane. Which method should I use? I know that the normal vector of my ...
2
votes
1answer
77 views

How To Generate Random Points on the Positive Side of a Plane in 3-D

Edit: The question can also be interpreted as: How to generate random coplanar points in a cube? Here is what I am struggling with: I have a cube, whose origin is $(0,0,0)$ and one edge length ...
1
vote
1answer
25 views

A rectangle $OACB$ with two axes as two sides,the origin $O$ as a vertex is drawn in which the length $OA$ is four times the width $OB$…

A rectangle $OACB$ with two axes as two sides,the origin $O$ as a vertex is drawn in which the length $OA$ is four times the width $OB$.A circle is drawn passing through the points BC and touching ...
1
vote
1answer
25 views

analytic geometry , Orthogonal projection

II) take the point P = (2,1,0) , the line r : X = (0,0,0) + t(2,1,0) and the plane B: x + y + z -3 = 0 . For each point Q let it be Q' his orthogonal projection to B .Find the Q points that outputs ...
1
vote
2answers
26 views

analytic geometry , perpendicular planes and a line

Find a equation to the plane that contain the line $X = (1,0,2) + t(4,1,0)$ and is perpendicular to the plane $A : 3x + y + z = 0$
1
vote
1answer
49 views

triangle proof: intersection of $w_\alpha$ and $m_a$ is outside $\Delta ABC$

I try to proove, that the intersection $M$ of $w_\alpha$ (angle bisector) and $m_a$ (perpendicular bisectors of $a$) of any triangle $\Delta ABC$ is always outside $\Delta ABC$, except $\Delta ABC$ is ...
1
vote
0answers
104 views

Computing Euler Angles from Direction Cosines Vector

My problem simply as the following: Suppose we measured the orientation of a plane object with respect to a reference fame. (where the reference frame parallel to plane frame). The unit normal vector ...
1
vote
0answers
28 views

Equations for Intersection of Plane and circle

I am having a problem in getting the related equations for an intersection between a point on a plane and the edge of a circle in 3D space. Any suggestions? Or also a tangent plane and a point on a ...
2
votes
0answers
55 views

A sphere packing problem

Suppose there is a large sphere of radius $R$. We want to pack it with smaller spheres. The volume of the smaller spheres change depending on where they are situated in the larger sphere. A smaller ...
0
votes
2answers
166 views

To prove by vector method , non-parallel sides of a trapezium having equal diagonals are equal.

How do we prove by vector method that "if the diagonals of a trapezium have equal length then the non-parallel sides of the trapezium have equal length." ? (taking $ABCD$ to be the trapezium with ...
1
vote
0answers
40 views

Determine properties of a shape defined by a function?

I have a number of functions that define 3d shapes. They take a point $(x, y, z)$ as a parameter and return a real value representing information about the shape. On the surface of the shape: $f(x) ...
2
votes
2answers
74 views

Is there a real continuous function of a Cube?

Sorry if this seems like a basic question, but I'm having troubles finding an answer in my research. I'm looking for a function that if given a test point (x, y, z) will return a real value describing ...
0
votes
1answer
49 views

How to find the locus that satisfy these conditions? [closed]

Find the locus of a point $P(x,y)$ such that the difference of its distances from $(3,0)$ and $(-3,0)$ is $2$.
0
votes
0answers
28 views

Area of intersection between square and annulus

The annulus' larger radius is $1$, smaller radius is $r>0$, and center is $(0,0)$. The square's sides are parallel with the axes, the lower left corner's coordinates are $(a,b)$, and the upper ...
1
vote
2answers
145 views

Books on geometric transformations and/or analytic geometry?

I've been looking to expand my knowledge in geometry as it's not covered in my undergraduate curriculum. For some reason I'm repelled by the classical approach (hopefully it will pass) as I feel it's ...
0
votes
5answers
78 views

Given Two Lines, How to Find a Point on Line $2$ given a Specific Distance From Line $1$?

I have two lines (U and V). What is the method to calculate a point on V given a specified distance (d) from U? The lines may be assumed that they do intersect (are not parallel) and are straight ...
4
votes
3answers
72 views

Proof that a line cuts in half the area of a parallelogram iff it goes through the intersection of the diagonals?

I read a theorem in a book which says that a line bisects a parallelogram iff it goes through the intersection of the diagonals. The edge case of this is of course if the line is one of the diagonal ...
2
votes
1answer
33 views

Finding the locus of the following conditions

Find the locus of all points in the plane, whose distance from a constant point $F=(x_0,y_0)$ divided by their distance from the vertical line $L=\{(k,y)\mid y \in \mathbb R \}$ equals a constant ...
1
vote
1answer
58 views

translate coordinates on circle to percentage?

I'm coming more from a programming point of view but the question is pure math. The only strange thing, I guess, is that the coordinate system is like this: ...
-1
votes
1answer
55 views

A problem on circles

Find the equation of the circles passing through two points on the $y$ axis at distances 3 units from the origin and having radius 5. (This a homework problem but I do not know how to solve it.)
1
vote
2answers
33 views

Calculating the coordinates of a vertex within a triangle so that the triangle becomes a right-angle

I have three points defining the vertices of a triangle A(1,4) B(6,7) C(5,1) I have found that vector AC has a slope(m) of 0.6 and vector BC has a slope of 6. From these slopes have the angles of ...
1
vote
1answer
90 views

Finding a polar equation for an ellipse

I need to find the polar equation of an ellipse, with one of its foci at the pole (origin), with a horizontal major axis of $10$ units and a vertical minor axis of $6$ units. So first off, the ...
2
votes
1answer
46 views

When are $ r_1: ax + by + c=0 $ and $ r_2: a'x + b'y + c'=0$ distinct?

When are $ r_1: ax + by + c=0 $ and $ r_2: a'x + b'y + c'=0$ distinct? I think, if $$rank\left(\begin{array}{cc} a & b \\ a' & b' \end{array} \right)\neq 1$$ then $r_1$ and $r_2$ are ...
0
votes
1answer
28 views

What type of Math and how to solve (points and distance)

I think it has to do with geometry and algebra. We have a rectangle with four points. We know each point as $(0,0), (0,10),(10,10)$ and $(10,0)$. In the rectangle we have another point $(x,y)$. ...
1
vote
0answers
37 views

Stereographic projection of the icosahedron and snub cube?

Using a steoreographic projection, the three equations associated with the icosahedron with unit circumradius, inradius, and midradius (respectively) are, $$f=z^{20} - 228z^{15} + 494z^{10} + 228z^5 ...
1
vote
1answer
50 views

Product of gradients of x=0 and y=0

A friend asked me this question: The product of the gradient of any two lines perpendicular to each other is $-1$. Now, the lines $x=0$ and $y=0$ are perpendicular to each other. If you take the ...
0
votes
1answer
33 views

A question of straight lines

If the straight lines $x+y-2=0$, $2x-y+1=0$ and $px+qy-r=0$ are concurrent, then what is the slope of the member of family of lines $2px+3qy+4r=0$ which is farthest from origin? I wrote the ...
0
votes
1answer
33 views

Fixed points through a general circle.

The circle $C: x^2 + y^2 + kx + (1+k)y - (k+1)=0$ passes through two fixed points for every real number $k$. Find $(i)$ co-ordinates of these two points and $(ii)$ the minimum value of the radius.
0
votes
1answer
122 views

how to find the barycentric coordinates of the orthocenter

$A = (0,0),B = (4,0),C = (1,2)$ How can I find the barycentric coordinates of the orthocenter of $\triangle ABC$?
0
votes
2answers
83 views

Find the equation defining a perpendicular bisector

Hello fellows, I've not had much time to post questions, but I post this one because while in my Maths lesson, I became annoyed by solving the same thing over and over again, when a good ...