Articles of geometry

Problem regarding the fitting cube into sphere

I was solving the following problem Suppose I have a sphere of radius 1 metre. The sphere is colored with red and blue such that it has disconnected regions of red and blue colors. Now I have to make a cube that fits in the sphere such that Each vertex of the cube touches a […]

Show that $PF.PG=b^2$ in a hyperbola

If the normal at P to the hyperbola $\frac {x^2}{a^2}-\frac {y^2}{b^2}=1$ meets the transverse axis in G and the conjugate axis in G’ and CF be the perpendicular to the normal from the center C then show that $$PF.PG=b^2\space and \space PF.PG’=a^2.$$ We know that the equation of the normal at parametric point $P\equiv(a\sec\theta,b\tan\theta)$ is […]

Curve of a fixed point of a conic compelled to pass through 2 points

Suppose that in the plane a given conic curve is compelled to pass through two fixed points of that plane. What are the curves covered by a fixed point of the conic, its center (for an ellipse), its focus, etc. ? (I apologize for the bad English …)

Bisectors of a triangle meet at point.

Prove that internal angle bisectors of $\triangle ABC$ meet at a point. The problem is that I have to prove this using the locus of a straight line and its properties, I can’t use vectors. The proof I can think of is very simple but extremely tedious. Let the coordinate of triangle be $(0,0), (a,0), […]

Computing a point having distances from 2 points respectively

I want to compute a point $p$ which has distances $d1$ and $d2$ from points $q1$ and $q2$ respectively. As I wanted a general answer, I used Maxima inputting the following script , $solve($ $[sqrt((num1-x)^2+(num2-y)^2)=num3,$ $sqrt((num4-x)^2+(num5-y)^2)=num6]$ $,[x,y]);$ Where all variables of the format $num$_ were constants in my mind and $p=(x,y)$. I expected an output […]

Fitting data to a portion of an ellipse or conic section

Is there a straightforward algorithm for fitting data to an ellipse or other conic section? The data generally only approximately fits a portion of the ellipse. I am looking for something that doesn’t involve a complicated iterative search, since this has to run at interactive speeds for data sets on the order of 100s of […]

Find the shortest distance from the triangle with with vertices in $(1,1,0),(3,3,1),(6,1,0)$ to the point $(9,5,0)$.

I have to find the shortest distance from the triangle with with vertices in $(1,1,0),(3,3,1),(6,1,0)$ to the point $(9,5,0)$. I cannot figure out how to do this. There are three possible cases: the minimum lies inside the triangle, on an edge or in a vertex. But what is a reasonable way to understand in which […]

The necessary and sufficient condition for a regular n-gon to be constructible by ruler and compass.

I have a problem concerning the necessary and sufficient condition for a regular n-gon to be constructible by ruler and compass. $\bf My$ $\bf question:$ For a given positive integer $n$, how can we prove that a regular $n$-gon is constructible by ruler and compass if and only if the number $cos(2\pi/n)$ (that is, the […]

Rigorous definition of “oriented line” in an Euclidean affine space

Let $\mathcal{A}^n$ be an affine space of dimension $n$. For example, let’s take $n=3$. A line $\mathcal{s}$ of $\mathcal{A}^3$ is an affine subspace of dimension $1$, that is: $\mathcal{s}=\{P \in \mathcal{A}_3 \text{ such that } \overrightarrow{AP} \in \langle u \rangle \}$. Now, what is not clear to me is: if we consider an Euclidean affine […]

How to calculate the two tangent points to a circle with radius R from two lines given by three points

I need to calculate the two tangent points of a circle with the radius $r$ and two lines given by three points $Q(x_0,y_0)$, $P(x_1,y_1)$ and $R(x_2,y_2)$. Sketch would explain the problem more. I need to find the tangent points $A(x_a,y_a)$ and $B(x_b,y_b)$. Note that the center of the circle is not given. Please help.