Questions on the use of algebraic techniques to prove geometric theorems.

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1answer
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Given the incentre of $\Delta ABC$ and the equations of the angle bisectors what is the locus of the centroid of the triangle $ABC$?

I got this problem on a test yesterday Consider $\Delta ABC$ with incenter $I(1,0)$. Equations of the straight lines $AI$, $BI$, and $CI$ are $x=1$, $y+1=x$ and $x+3y=1$ respectively and $\cot \left( ...
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5answers
355 views

Creative way to find this area

Let's say We have a circle with center at $(0,0)$ with radius $r$ and we have the line $y=a$ where $0 \leq a \leq r$. the question is what is the area that between the circle and the line $y=a$(the ...
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0answers
18 views

On the solutions of a system of inequalities avoiding Helly's theorem

Let $a_1,b_1,\cdots,a_4,b_4\in\mathbb{R},r_1,\cdots,r_4\in(0,+\infty)$. Show that, if $\not\exists (x,y)\in\mathbb{R}^2$ such that $$ \begin{cases} (x-a_1)^2+(y-b_1)^2\le r_1\\ ...
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1answer
103 views

Largest of the smallest angles of incidence from arbitrary point to tetrahedron vertex/centroid line

Picture a regular tetrahedron where each vertex has a line through the centroid and a plane normal to it. I need to show that the range of the smallest angles of incidence from an arbitrary point to ...
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1answer
21 views

Finding the equation of the new plane after the original has been rotated by an angle

Find the equation of the plane obtained after rotating the plane $x+y+z=1$ by $90^{\circ}$ about its line of intersection with the plane $x-2y+3z=0$. Since I had to choose one of the four given ...
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2answers
77 views

How to find center of a conic section from the equation?

If we are given a curve in the form $$ax^2+2bxy+cy^2+2dx+2ey+f=0$$ and the following determinant $$\delta=\begin{vmatrix}a&b\\b&c\end{vmatrix}=ac-b^2$$ is non-zero, then this is either a curve ...
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2answers
58 views

Compute Speed of an object when moving on a circular arc? [on hold]

Consider the following figure: An iPhon is moving on a circular arc from point A to point B. The radius of the orbit is f. Consider the case that a men stands and holding his arm horizontal to the ...
10
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1answer
92 views

Something Isn't Right With My Parking

A few days ago in my Calculus BC class we were given a page of 6 challenging end of the year problems. That was a refreshing change from the drudgery we usually do (WebAssign). One of them went like ...
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2answers
38 views

Find centre of circle with equation of tangent given

(4,1) is a point on one end of the diameter of a circle and the tangent through the other end of the diameter has equation 3 x- y=1. Determine the coordinates of the center of circle. What got me ...
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2answers
94 views

The Efficiency of Random Parking Problem

A few days ago in my Calculus BC class we were given a page of 6 challenging end of the year problems. That was a refreshing change from the drudgery we usually do (WebAssign). One of them went like ...
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4answers
413 views

Some theorems in euclidean geometry have incomplete proofs

I have seen that, in euclidean geometry, proofs of some theorems use one instance of the 'geometric shape'(on which the theorem is based) to proof the theorem. Like, the proof of 'A straight line ...
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2answers
5k views

The shortest distance between any two distinct points is the line segment joining them.How can I see why this is true?

On a euclidean plane, the shortest distance between any two distinct points is the line segment joining them. How can I see why this is true?
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4answers
2k views

What is a point?

In geometry, what is a point? I have seen Euclid's definition and definitions in some text books. Nowhere have I found a complete notion. And then I made a definition out from everything that I know ...
0
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1answer
33 views

On the definition of sphere in analytic geometry…

Last year, when I was teaching mathematics (analytic geometry) for one of my clever freands, I arrived to the definition of sphere. I said Fix $r>0$, An sphere is the set of all triples ...
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2answers
22 views

Find the slope of a line $L$ that tangent to the graph of $y = x^3$ and passes through the point $(0,2000)$.

Find the slope of a line $L$ that tangent to the graph of $y = x^3$ and passes through the point $(0,2000)$. Well, I am new to this concept, to me, slope means $\dfrac{dy}{dx}$, but I get ...
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4answers
269 views

Is there a name for the curve $t \mapsto (t,t^2,t^3)$?

Is there a name for the curve given by the parametrization $\{(t,t^2,t^3); t\in\mathbb R\}$? Here is a plot from WA. An another plot for $t$ from $0$ to $1$. This curve is an example of a ...
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2answers
52 views

Let $y=x^2+ax+b$ cuts the coordinate axes at three distinct points. Show that the circle passing through these 3 points also passes through $(0,1)$.

Let $y=x^2+ax+b$ be a parabola that cuts the coordinate axes at three distinct points. Show that the circle passing through these three points also passes through $(0,1)$. Since, the graph of the ...
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2answers
38 views

How can I find the coordinates of a point which is the reflection of a point about a line in 3D

I am currently working on a project on Matlab and I need to find the coordinates of a point which is reflected about a line. I know how to do it in 2D but in 3D things are getting ugly. So, we have a ...
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1answer
466 views

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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2answers
101 views

Rigorous books on geometry

I am looking for a rigorous book on both 2d and 3d euclidean geometry, and also how analytic geometry can be developed from synthetic geometry. I haven't really found such a book yet. I would be very ...
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1answer
37 views

Conjugate Hyperbolas.

What would be a good approach to tackle this problem. In a previous assignment I managed to show Pq=Pr. How do I show that this tangent intersects the conjugate hyperbola. Should I start by ...
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0answers
44 views

Analytic-geometrical properties of dodecahedron

Consider the following projection of a dodecahedron: An equilateral triangle can be projected to make points $A, B, C, D, E, F$ intersect with it's edges. What would be the mathematical proof (if ...
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7answers
879 views

Perpendicular bisector

"Show that BE is the perpendicular bisector to AC" I tried to prove this through Pythagoras, but the answer I got did not prove it was at a right angle, and therefore said it was not the ...
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1answer
567 views

Collective Term for $XY, YZ$ and $ZX$ Planes

Is there a collective term for the $XY, YZ$ and $ZX$ planes in $3D$ co-ordinate geometry? I was thinking "principal planes" but I'm not sure where I heard that.
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1answer
25 views

Analytic geometry, distances

Find the equation of the geometric place: Whose distance to the point $(4,0)$ equals half the distance to the straight line $x=19$ Im using the formula for distance between points $P(4,0), Q(19,0)$ ...
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4answers
35 views

Finding horizontal tangents to a function.

Find the points at which the line tangent to the following function is horizontal $$q(x)=(x+3)^4(2x-1)^7$$ Every time I've gotten to the point of finding $x$ the numbers are all irrationally too ...
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0answers
29 views

Reference request: history of analytic geometry

I am searching a book in the domain of the history of math, that describes the historical origins of analytic geometry, starting from Descartes (?), and that describes also its development (e.g. the ...
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0answers
8 views

analytic geometry question involving perpendicular vectors

Determine the parametrics equations of the straight line that passes by $A(-1,4,5)$ and is perpendicular to $r:P=(-2,1,1)+t(1,-1,1)$. Someone can solve this? I'm trying for more than a hour and I'm ...
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0answers
25 views

Intersection of a curve with a complex line

Given: $$ \left\{\begin{matrix}t =\frac{1}{n}\sqrt{n^{4}-z^{2} } & \\ z=im & \end{matrix}\right.$$ with $n<m$, positive integers (and $i$ the imaginary unit), if one wanted to ...
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1answer
351 views

$2D$ Line Segment - Triangle Intersection

I've seen similar questions but could not solve my problem with those. My question is how to detect an intersection of a line segment and a triangle on a 2D coordinate system? I don't need the point ...
3
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2answers
108 views

snugly fitted spheres in a cube

A larger sphere A, having a radius $R$ is snugly fitted in a cube (i.e. sphere A touches all six faces of the cube). Further, a small sphere B is snugly fitted in the corner of cube (i.e. sphere B ...
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1answer
27 views

Definition of angle between non-differentiable curves

(Background: I am trying to understand the definition of angle-preserving function..I posted a question earlier but I still have doubts) My question is:how is the angle between two curves defined if ...
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5answers
7k views

Finding an equation of circle which passes through three points

How to find the equation of a circle which passes through these points $(5,10), (-5,0),(9,-6)$ using the formula $(x-q)^2 + (y-p)^2 = r^2$. I know i need to use that formula but have no idea how to ...
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2answers
58 views

Easiest way to verify that $4x^2+y^2=1$ is an ellipse?

Normally I would just divide both sides by the number $4$ because it's not good in there, but I can't do it for $$4x^2+y^2=1$$ I must have $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ So what's the ...
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0answers
30 views

Using axis coordination to represent rotation matrix instead of angles

Euler angles give us clear matrix for conversion of a vector from car reference $Fr^C$ to earth reference $Fr^E$. If $\vec V$ is a vector in different frames it is represented differently: $$\vec ...
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2answers
18 views

Analytic geometry and definite integrals problem…

So, here's the problem: We have a parabola $y^2=2px$ and a line which is perpendicular to parabola and forms the angle $\frac{3\pi}{4}$ with x axis. I have to find the area between the parabola and ...
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0answers
26 views

Given five points and a line find the points of the line that lie in the conic through the five points [closed]

So I'm given 5 points in general position and a line, I already know the method using Pascal's theorem to find points in the conic but I dont know how to find specifically the ones that lie on a given ...
0
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1answer
407 views

Shortest distance between parallel line and plane

I've been doing questions regarding the shortest distance between lines/planes and points , and I've come across a question asking to find the shortest distance between a line and a plane which are ...
1
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1answer
479 views

How to show that a line pass through a point?

How to show that a line pass through a point? Two fixed straight line $OX$ and $OY$ are cut by a variable line at the points $A$ and $B$ respectively and $P$ and $Q$ are the feet of the ...
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1answer
51 views

High School Geometry problem with a triangle and trapezoid in a larger triangle.

In school, I have an assignment to write a problem for geometry students. I have written the following problem. Draw triangle ABC. Let the height have magnitude h. Draw a line segment, DE, which is ...
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2answers
20 views

Characterize a rotation matrix

Given a matrix $A\in M_{2 \times 2}(\mathbb R)$ or $M_{3\times 3}(\mathbb R)$ how to determine if it is a rotation matrix? Is there any theorem that characterize a rotation matrix just by looking at ...
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3answers
55 views

Coordinate Geometry - Area of a Quadrilateral

What is the area in square units, of a quadrilateral whose vertices are $(5,3), (6,-4), (-3,-2), (-4,7)$ ? I have tried creating the triangles, but didn't know how to find the diagonal. I wanted to ...
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1answer
55 views

Is there a problem in assuming that a point is the same thing of a vector?

I've read Apostol's Calculus, in the section on analytic geometry. He says that he's going to use 'vector' and 'point' interchangeably. But in Beardon's Algebra and Geometry, he argues that there is ...
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2answers
3k views

rotation by 180 angle

in general i know that if we rotate (x,y) about origin by the 180 degree we will get new image (-x,-y),but suppose that we make rotation not about origin but some other point (a,b) does your result ...
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0answers
33 views

Analytic Geometry - vectors and points

Can somebody help me? In the picture, $\|AM\|=2\|MB\|$ and $\|AN\|=\frac{1}{3}\|CN\|$. Write $X$ in function of $A, AB, AC$.
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3answers
118 views

How can the centers of these 5 related circles be specified as a formula?

This is my first time posting in this forum, so please forgive me if my question is too involved or if I've posted it in the wrong area. I hope someone finds it interesting enough to try their hand at ...
1
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1answer
36 views

Hyperplanes divide space

Problem. What is maximal number of connected components on which $n$ hyperplanes divides $\mathbb{R}^m$ if they all have 1 common point. In fact this problem was firstly stated in $\mathbb{R}^3$ and ...
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2answers
52 views

Surface area of a section of the unit sphere

Let $v$ be a vector on the unit sphere in $\mathbb{R}^n$ and let $S(\epsilon)$ be the set of vectors $s$ on the same sphere such that $$ |s \cdot v| \leq \epsilon.$$ What is the surface area of ...
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3answers
2k views

Fascinating Lampshade Geometry

Today, I encountered a rather fascinating problem in a waiting room: Notice how the light is being cast on the wall? There is a curve that defines the boundary between light and shadow. In my ...
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2answers
319 views

Circle touching the $y$-axis passing through two points

How to find the equation of the circle touching the $y$-axis given that it passes through two particular points?