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

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42
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4answers
9k views

False proof: $\pi = 4$, but why?

Note: Over the course of this summer, I have taken both Geometry and Precalculus, and I am very excited to be taking Calculus 1 next year (Sophomore for me). In this question, I will use things that I ...
0
votes
1answer
20 views

How to find a transversal of two lines that is also perpendicular to a plane

I have this problem where I have two lines given and I have to find a transversal. However, it also has to be perpendicular to a given plane (lines are not necessarily in the given plane). My guess ...
0
votes
0answers
13 views

Does a tangent exist at $x=0$ to $y=sgn(x)$?

Yesterday my professor told me that a tangent can be constructed at $x=0$ to the signum function reasoning that the two points considered while drawing a tangent must be close horizontally and not ...
1
vote
1answer
44 views

Calculate point of intersection line of two planes

I found some source code that I do not really understand. I will give some pseudo-code in my description to give you a better idea how the algorithm works. Basically, two planes with three vertices ...
0
votes
0answers
27 views

. Find the projection of the triangle on the coordinate planes.

Given the following, three vectors: a⃗ =3i−2j+5k b⃗ =i−6j+6k c⃗ =2i+3j−k Relative to cartesian coordinate systems with origin O. I calculated the sides to be 4.58,11.45 and 7.87. I also calculated ...
0
votes
1answer
18 views

Equation of parabola from 2 points and an angle at the first point

A parabola starts at a coordinate A and ends at coordinate B. Angel of tangent through A is given 'theta'. With these data (A,B,theta) how can I get an equation of a parabola?
0
votes
0answers
13 views

General formula or at least check of existence of some given n-1 polytope cross section obtain by one cut on a given n polytope?

This is a curious observation inspired after beating this game So for these two levels in question, after some prior experience in common cross sections and loads of trial and error, I found it is ...
1
vote
1answer
33 views

How to find the length of a line segment inside of the unit cube?

$$\frac{x}2=y=\frac{z}{-2}$$ What is the length of the segment inside of the unit cube? I guess I should find the intersections of the line and the $x=1,y=1,z=1$ planes but I think this line ...
1
vote
3answers
42 views

Orthocenter of triangle $DEF$ is same as the circumcenter of triangle $ABC$

$D,E,F$ are mid points of the sides of the triangle $ABC$,then prove that the orthocenter of triangle DEF is same as the circumcenter of triangle ABC. I cannnot figure out what coordinates to suppose ...
1
vote
0answers
37 views

$1 \leq a^2x^2 + b^2y^2 - abxy \leq 9 , x\geq 1$- question compactness and connectedness..

I was told that this object was a cone, I cannot see that, can anyone tell me how to identify which object this is, so as to continue assesing and answering questions of compactness and ...
0
votes
2answers
27 views

Applications of derivatives Analytical geometry

Any tangent at a point $P (x,y)$ to the ellipse $x^2/8 + y^2/18 =1$ meets the coordinate axes in the points $A$ and $B$ such that the the area of triangle $OAB$ is least where $O$ is the origin. Then ...
0
votes
1answer
19 views

If the tangents at

If the tangents at $P(1,1)$ on the curve $y^2 =x(2-x)^2$ meets the curve again at $Q$ then points of $Q$ is of the form $(3a/b,\, a/2b)$ so I have to find $a$ and $b$.
1
vote
2answers
76 views

Points of intersection of two lines

I have two lines that are concurrents and I want to know the point of intersection between them. To find the point my algorithm performs the following equation and replacing the lambda found in one of ...
1
vote
1answer
28 views

Coincidents lines

My algorithm to determine whether two lines are coincident (which have been proven previously they are parallel) verifies the following equation: $$ \dfrac{x - x_o}{a} = \dfrac{y - y_o}{b} = \dfrac{z ...
1
vote
1answer
40 views

Intersection point of two lines in 3D

I need an algorithm that returns the point of intersection between two lines. The algorithm is capable of determining the relative position then I'm sure the lines will intersect. My question is: I ...
0
votes
4answers
97 views

Proof of Pythagorean theorem without using geometry for a high school student?

There are some proofs of Pythagoras theorem which don't even require high school maths to understand, but they all are using shapes to prove of the theorem. However, I am trying to find some proofs of ...
0
votes
1answer
21 views

Equation of BI,CI given and angle A to be found

If $I(1,0)$ is the center of incircle of triangle ABC,the equation of BI is $x-1=0$ and the equation of CI is $x-y-1=0$,then angle BAC is (A)$\frac{\pi}{4}$ (B)$\frac{\pi}{3}$ (C)$\frac{\pi}{2}$ ...
6
votes
4answers
96 views

Area of a triangle with sides $\sqrt{x^2+y^2}$,$\sqrt{y^2+z^2}$,$\sqrt{z^2+x^2}$

Sides of a triangle ABC are $\sqrt{x^2+y^2}$,$\sqrt{y^2+z^2}$ and $\sqrt{z^2+x^2}$ where x,y,z are non-zero real numbers,then area of triangle ABC is (A)$\frac{1}{2}\sqrt{x^2y^2+y^2z^2+z^2x^2}$ ...
1
vote
2answers
27 views

Two circles touching a line and the axes

If the circle $C_1$ touches x-axis and the line $y=x \tan\theta$,$\theta \in (0,\frac{\pi}{2})$ in the first quadrant and circle $C_2$ touches the line $y=x \tan\theta$,y-axis and circle $C_1$ in such ...
1
vote
1answer
23 views

Distance between incenters and excenters

In a triangle ABC,if $(II_1)^2+(I_2I_3)^2=\lambda R^2$,where I denotes incenter,$I_1,I_2,I_3$denotes centers of the circles escribed to the sides BC,CA and AB respectively and R be the radius of the ...
0
votes
0answers
79 views

The maximum value of PA.PB.PC

Let A,B,C be the vertices of a triangle inscribed in a unit circle, and let P be a point in the interior or on the sides of the triangle ABC. Then the maximum value of (PA)(PB)(PC) equals to? I could ...
0
votes
1answer
30 views

Range of slope of line PQ

Let $A(-1,0),B(3,0)$ and PQ be any line passing through (4,1).The range of the slope of PQ for which there are two points on PQ at which AB subtends a right angle is $(\lambda_1,\lambda_2)$,then what ...
1
vote
1answer
21 views

Sorting triangles by hypotenuse length

I have some points in $xy$ space and I need to sort distances between these points. If I calculate real distance, then I need to perform $\sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}$ and this is very time ...
0
votes
1answer
67 views

Intersection of Three Planes proof

I'm supposed to be making a study guide answer for this question, but I'm struggling with proof. Show that the three planes intersect at the point provided that Note that the ...
3
votes
3answers
49 views

Find the equation of a circle that intersects the $y$-axis at the origin and at the point $(0,6)$ and also touches the $x$ axis. - basic question

Find the equation of a circle that intersects the $y$-axis at the origin and at the point $(0,6)$ and also touches the $x$ axis. Okay, so I wasn't sure how to do this so I looked at the answer at ...
1
vote
1answer
18 views

Angle bisector equation and orthocenter given, vertex to be found

In triangle $ABC$, let $A(3,4)$ and the equation of angle bisector of $B$ is $y=x$. If orthocenter of triangle is $(2,2)$ and $B(h,k)$, then find $(h,k)$. I cant solve this question, I am confused, ...
1
vote
2answers
42 views

Question based of orthocenter distance from angular points

In an acute angled triangle ABC,$\angle A=20^\circ $,let D,E,F be the feet of altitudes through A,B,C respectively and H is the orthocenter of $\bigtriangleup ABC $.Find ...
1
vote
1answer
31 views

Polarity on a Hyperboloid of one sheet

Given a quadric $Q = \{v \in \mathbb{R}^n \mid \alpha(v,v) = 1\} \subset \mathbb{R}^n$, defined by a bilinear form $\alpha: \mathbb{R}^n\times\mathbb{R}^n \to \mathbb{R}^n$, and an affine subspace ...
1
vote
1answer
42 views

Equation by Graph

Given a random Graph,is there any known way to find an equation for it ? If I create a random graph is there a way that i can find an equation that totally describes my random graph?
1
vote
2answers
37 views

Angle bisector related question

The internal bisectors of the angles of a triangle ABC meet the sides in D,E,and F.Show that area of the triangle DEF is equal to $\frac{2\Delta\times abc}{(b+c)(c+a)(a+b)}$,here $\Delta $is area of ...
3
votes
2answers
61 views

Finding the equation of a circle.

A circle of radius $2$ lies in the first quadrant touching both axis. Find the equation of the circle centered at $(6,5)$ and touching the above circle externally. Let me share how I answered this ...
2
votes
5answers
74 views

Find the equation of the circle.

Find the equation of the circle whose radius is $5$ which touches the circle $x^2 + y^2 - 2x -4y - 20 = 0$ externally at the point $(5,5)$
0
votes
3answers
32 views

Derivation of the equation for the envelope

Suppose we have a family of curves on the plane. The equation of the curves is given by $$ f(x ,y ;t) = 0 . $$ Here $t$ is the parameter. On Wiki, the equations determining the envelope of this ...
0
votes
1answer
16 views

$|\vec{r}-\vec{r}_1|=\frac{1}{2}|\vec{r}-\vec{r}_2|$ is the equation of a sphere?

I am told that the set of all $\vec{r}$ for which $|\vec{r}-\vec{r}_1|=\frac{1}{2}|\vec{r}-\vec{r}_2|$ is true forms a sphere---however, my semi-intuitive reading of this equation puts a "weaker ...
1
vote
1answer
20 views

Orthocenter and coordinates of a vertex

In a triangle ABC,the vertex A is (1,1) and orthocenter is (2,4).If the sides AB and BC are members of the family of straight lines $ax+by+c=0$,where $a,b,c$ are in arithmatic progression.Then the ...
1
vote
0answers
27 views

Disk in analytic topology vs. the spectrum of a Henselian DVR in etale topology

In this informative and concise set of notes on vanishing cycles by Donu Arapura, it is stated that the theory of vanishing cycles ports nicely to the etale world if the role of the disk is replaced ...
0
votes
1answer
25 views

Equivalence of euclidean and analytic geometry [closed]

I read about the axioms of euclidean geometry. How is analytic geometry rigorously defined? What are the axioms? And most important: How to prove that all the results proved in analytic geometry are ...
1
vote
2answers
44 views

Minimum value of an an expression

Find the minimum value of $(\alpha-\beta)^2+(\sqrt{2-\alpha^2}-\frac{9}{\beta})^2$ where $ 0<\alpha<\sqrt{2}$ and $\beta>0$ My attempt: In my view,this minimum value is the shortest ...
0
votes
2answers
15 views

How do I find intersections between a circumference and an equilateral hyperbola?

Let's say I have a circumference with the equation $x^2 + y^2-10=0$. This circumference has a point A $(1;3)$ which which passes thorough an equilateral hyperbola $xy=3$. I would like to find all the ...
0
votes
2answers
21 views

Find the equation of an equilateral hyperbola passing through a point of a circumference

Let's say I have a circumference with the following equation $x^2+y^2-10=0$, the coordinates of its center are $(0;0)$ and its radius is $\sqrt{10}$. I need to find the equation a equilateral ...
5
votes
2answers
45 views

Find a plane with distance $3$ from $3x-y-z = 0$

I need to find a plane such that its distance from the plane $3x-y-z = 0$ is $3$. Since distance is defined only for parallel planes, I already know that they have to be parallel, and then, the ...
-1
votes
4answers
55 views

Find the distance between the point $(0,0,0)$ and the plane $2x+3y+z=1$ [closed]

Find the distance between the point $(0,0,0)$ and the plane $2x+3y+z=1$. So I know in order to find the distance I need two points. How do I find a point in the plane?
0
votes
2answers
39 views

Find the cartesian equation of the locus of the set of points of $P$ problem.

Find the cartesian equation of the locus of the set of points of $P$. $P$ is at a constant distance of five units from the line $4x-3y=1$ I don't have much intuition on how to solve this one. ...
2
votes
1answer
67 views

Point on the Plane, a Triangle, and a Lower Bound of a Ratio Sum

Let $ABC$ be a triangle on the Euclidean plane. At which point $P$ on the plane does the ratio sum $\frac{PA}{BC}+\frac{PB}{CA}+\frac{PC}{AB}$ attain its minimum value? Prove also that, for any ...
-1
votes
1answer
41 views

Can anyone help me with this contradicting graphs?

While studying SAT MATH 2 , I tried to solve the following problem but faced some difficulty. The problem goes ........ In the graph of the parametric equations $x= t^2+t$ , $y=t^2-t$ A) $x\ge 0$ ...
3
votes
1answer
81 views

reference on $\sqrt{ax}+\sqrt{by}=c$ as a parabola?

Does anyone have a reference on the equation $$\sqrt{ax}\,+\sqrt{by}=c\ ?$$ Clearing square roots and rearranging gives $$ax+by = \frac{(ax-by)^2+c^4}{2c^2}$$ This is the equation of a parabola, so ...
0
votes
1answer
43 views

differentiating an integral with respect to a variable which also affects the region of integration

I am considering taking the derivative of the function $$F(\mathbb{x_1},\mathbb{x_2},\mathbb{x_3}) = \displaystyle \int_{V_1} ||x-\mathbb{x_1}||\phi(x)\,dx + \int_{V_2} ||x-\mathbb{x_2}||\phi(x)\,dx ...
2
votes
3answers
64 views

Drawing circumference issue

I'm a developer, and I'm developing an app on Google Maps. At the moment, I'm trying to draw a circle on the map. For getting all the points I need, I'm using the following formula: \begin{equation} ...
1
vote
1answer
33 views

Interpolating Random Points

I have a list of (x,y) co-ordinates that need to be interpolated. The co-ordinates are not necessarily part of a function. Therefore, polynomial interpolation will not work. Is there a way to use some ...
3
votes
1answer
65 views

Rotation of complex numbers in a complex plane. Check my work?

Say that $c_1 = -i$ and $c_2 = 3$. For this problem, let $z_0$ be an arbitrary complex number. We can rotate $z_0$ around $c_1$ by $\pi/4$ counterclockwise to get $z_1$. Next, we canrotate $z_1$ ...