# Tagged Questions

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### n-gon Inequality Theorem Converse

In the plane, if we have an n-gon with side lengths $v_1$, $\ldots$,$v_n$, these lengths satisfy the "planar $n$-gon inequalities," ie. the length of each side is less than the sum of the lengths of ...
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### 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 $\sin^2 A = \sin^2 B+ \sin^2 C - 2\sin B \sin C \cos A$
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### find the equation of the locus of a moving point which is always equidistant from the y-axis and the point (-6,4)

Do you know how to solve its equation? Already solved some locus problems that gives points but not in the y or x axis problems.
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### 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 ...
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### 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 ...
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### $3x+3y-1,4x^2+y-5,4x+2y$ are sides of an equilateral triangle

I am completely lost in this one $3x+3y-1,4x^2+y-5,4x+2y$ are sides of an equilateral triangle, its area is closest to the which integer?
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### Connecting Coordinate Geometry and Plane Geometry

What is it that allows us to take theorems proven in Euclidean geometry (i.e. with Euclid's five postulates or Hilbert's Axioms) and then apply them outside of Euclidean geometry. For example in ...
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### Is it true that all of the euclidean geometry problem in the IMO(international mathematical olympiad) could all be solve by the analytical geometry?

Is it true that all of the euclidean geometry problem in the IMO(international mathematical olympiad) or even generalize to say that all the plane geometry problem and 3d-geometry could be solve by ...
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### How to find the points of tangency of a parabola using Calculus?

How can someone find the points of tangency of a parabola in this situation? I need to find two points of tangency so that the triangle formed by the two tangent lines at those points and the x axis ...
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### Find locus of points relating to an ellipse

I would like to find the equation of the following locus. For a big circle C centered at (0,0), the locus of points that the sum of distances to Y-axis and to C is 1, say in the first quadrant, is ...
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### Why do we believe the equation $ax+by+c=0$ represents a line?

I'm going for quite a weird question here. As we know, the equation in Cartesian coordinates for a line in 2-dimensional Euclidean geometry is of the form $ax+by+c=0$. I'm wondering why do we ...
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### Two points: Distance, midpoint, equations of line passing through them and perpendicular line [closed]

Consider two points (-2,3) and (-1,6). Calculate the distance between these two points. Find the midpoint. Obtain the equation of the line passing through the given points. Find the equation of the ...
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### how do i calculate coordinates of point on rectangle based on angle

If i want to find out what is x and y of point lies of a line drawn from center of the rectangle to outward at certain angle. take a look at picture below.
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### How do I move through an arc between two specific points?

I've found many answers to similar questions here, but I'm still stuck. I want to move an object from point sx,sy to point dx,dy through an arc that bulges by distance b from the line straight ...
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### Equation of a line passing through a given point, perpendicular with a vector

Find the line that goes through A(1,0,2) and is perpendicular to r = (-2,3,4) + s (1,1,2) I did a bunch of work, but I don't know if any of it is right. I erased most of it, but this is what I came ...
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### 3D Geometry Question

In $3$-dimensional Geometry, if angle made of line segment $OP$ with $X,Y,Z$-axis are in $1:2:3$, then what is the angle made by line segment with $Y$-axis? My Solution: Let $\alpha,\beta$ and ...
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### closest point to on $y=1/x$ to a given point

I feel like I'm missing something basic - given a point $(a,b)$ how do I find the closest point to it on the curve $y=1/x$? I tried the direct approach of pluggin in $y=1/x$ into the distance formula ...
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### Is the area of intersection of convex polygons always convex?

I am interested specifically in the intersection of triangles but I think this is true of all convex polygons am I correct? Also is the largest possible inscribed triangle of a convex polygon always ...
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### How to find the intersection of the area of multiple triangles

I have a couple of questions regarding finding the intersection of triangles. I have a system of 16 projectors that all have slightly different color gamuts. The color gamuts are represented by a ...
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### Faster Distance formulae for higher n dimension

I need to calculate the distance two points, but these two points contain more than 100 dimensions. With the regular two or three dimension distance formula, we can extend this further to n dimension ...
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### Number of ellipses through two fixed points in 2D space?

How many ellipses with a given size (mean $a$ and $b$ given) one can draw through two fixed points in 2D plane?
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### Locus perpendicular to a plane in $\mathcal{R}^4$

I have solved an exercise but I'm not sure to have solved it perfectly. Could you check it? It's very important for me.. In $\mathcal{R}^4$ I have a plane $\pi$ and a point P. I have to find the ...
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### Equations of branches of a mind map

Sorry for the long question, but it's not so simple to explain. Consider a mind map like this: I want to draw branches in a cartesian coordinate system. I'd like to find two equations which ...
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### A curve that intersects every plane in finitely but arbitrarily many points

Does there exist a piecewise smooth curve in $\mathbb{R}^3$ such that every plane intersects the curve at finitely many points and the number of intersection points can be arbitrary large? If the ...
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### Figure out if a fourth point resides within an angle created by three other points

If I have a point that is considered the origin and two lines that extend outwards infinitely to two other points, what is the best way to determine whether or not a fourth point resides on or within ...
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### Geometry IMO 1988

(IMO 1988/1) Consider two circles of radii $R$ and $r$ $(R > r)$ with the same center. Let $P$ be a ﬁxed point on the smaller circle and $B$ a variable point on the larger circle. The line $BP$ ...
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### Find the area of overlap of two triangles

Suppose we are given two triangles $ABC$ and $DEF$. We can assume nothing about them other than that they are in the same plane. The triangles may or may not overlap. I want to algorithmically ...
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### How to find the interior angle of an irregular pentagon or polygon?

I have 5 points and measures of sides of pentagon in 2D. Then how do i find interior angles of pentagon? Suppose $P_1,P_2,P_3,P_4,P_5$ are five points of Pentagon $P_1P_2P_3P_4P_5$. I know how to ...
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### Prove a very original version of Descartes's circle theorem

Prove: I define the radius of three mutually externally tangent to be $d,e,f$ respectively. The circle with radius $x$ is internally tangent to all three circles. Then ddeeff+ddeexx+ddffxx+eeffxx ...
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### Determinant and Measure

The determinant of the matrix of its vectors gives the measure of an $n$-dimensional parallelogram. For example, in $2$ dimensions, the area spanned by vectors $v$ and $w$ is \begin{array}{|cc|} v_1 ...
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### Can more than four circles internally tangent or external tangent or combination of both each others at different points?

Is it true for infinite number of m, more than four, there exist m circles internally tangent or external tangent or combination of both each others(in this problem, i mean a circle must be tangent to ...
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### Finding unknown 3D vector given 2 known vectors and 2 known angles

So I have a 3D vector math problem that I'm having difficulty solving. Basically I have two known vectors in the form (x,y,z), let's call them C and P, and I want to find a third unknown vector, let's ...
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### What is wrong with this proof that isometries must be surjective?

Let $\phi : \mathbb{R}^2\rightarrow\mathbb{R}^2$ be an isometry. Suppose $\phi$ is not surjective, that is there exists some $v \in \mathbb{R}^2$ whose fiber $\phi^{-1}(v)$ is empty. Then by the ...
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### Is there a generalized method of rotation for curves?

I know that we can rotate a curve in $R^2$ about a linear axis, as is common for first year calculus problems involving solids of revolution. But has anyone come up with a general method to take a ...
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### Proving two lines trisects a line

A question from my vector calculus assignment. Geometry, anything visual, is by far my weakest area. I've been literally staring at this question for hours in frustrations and I give up (and I do mean ...
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### Determining if an arbitrary point lies inside a triangle defined by three points?

Is there an algorithm available to determine if a point P lies inside a triangle ABC defined as three points A, B, and C? (The three line segments of the triangle can be determined as well as the ...
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### Locus and concurrent lines

This will be my first question :-) Let $\mathcal{D}_1$ and $\mathcal{D}_2$ two concurrent lines, and $F$ a point in the plane, and $H$ and $G$ its images by the symmetries of axis $\mathcal{D}_1$ and ...
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### Distribution or bounds for maximum Cartesian coordinate sampled from the sufarce of an n-sphere

It's been said that for high dimensions a hypersphere is "nearly all equator". The amount of space near the poles is just ridiculously small. This of course means that from a uniformly random sample ...
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### Finding a random vector exactly yay far from another point in 3D space

So I am trying to find a vector a certain distance away from another point ( the distance varies based on an input ) and I've figured out that ...
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### Product of slopes is -1 iff perpendicular proof from first principles

Once again I'm working through Stillwell's Four Pillars of Geometry. I'm on Chapter 3 where he first introduces coordinates. The question reads, 3.5.1 Show that lines of slopes $t_1$ and $t_2$ ...
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### Is this lemma about the minimal distance of two lines true?

In school, I recently proved a solid geometry excercise by assuming that the following lemma is true: If two lines $g$ and $h$ in the euclidian space are not parallel, and if the lines seem ...