Articles of geometry

Finding the number of normals to a parabola

Find the number of normals to the parabola $y^2=8x$ through (2,1) $$$$ I tried as follows: Any normal to the parabola will be of the form $$y=mx-am^3-2am$$ Since the point (2,1) lies on the normal, it satisfies the equation of the normal. Thus $$2m^3+2m+1=0$$ The number of real roots of $m$ in the above equation […]

What is the probability that the resulting four line segments are the sides of a quadrilateral?

Question: Divide a given line segment into two other line segments. Then, cut each of these new line segments into two more line segments. What is the probability that the resulting line segments are the sides of a quadrilateral? I am stuck on this problem. I think I am close, but I am not sure […]

Pullback Calculation

If we define the 2-form $\omega=\frac{1}{r^3}(x_1dx_2\wedge dx_3+x_2dx_3\wedge dx_1+x_3dx_1\wedge dx_2)$ with $r=\sqrt{x_1^2+x_2^2+x_3^2}$ If we now define $x(\theta,\phi)=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$ for $\theta\in(0,\pi)$ and $\phi\in (0,2\pi)$ I now want to show that the pullback of this map is: $x^*\omega=\sin\theta d\theta\wedge d\phi$ Now my definition of the pullback of a map $c:[a,b]\rightarrow D$ is $\alpha(c'(t))dt$ but am unsure as to how […]

Prove the inequality $4S \sqrt{3}\le a^2+b^2+c^2$

Let a,b,c be the lengths of a triangle, S – the area of the triangle. Prove that $$4S \sqrt{3}\le a^2+b^2+c^2$$

Second point of intersection of two circles in Möbius geometry

Suppose I have two circles \begin{align*} (x-x_1)^2+(y-y_1)^2 &= r_1^2 \\ (x-x_2)^2+(y-y_2)^2 &= r_2^2 \end{align*} and I also have one point $$p=(x_p,y_p)$$ which is known to lie on both of these. How can I find the other point of intersection most elegantly? In particular, I’d like to avoid case distinctions, so I need an implementation which […]

Have you seen this golden ratio construction before? Three squares (or just two) and circle. Geogebra gives PHI or 1.6180.. exactly

Note this golden ratio construction has been dramatically updated here with numerous golden harmonies: A Golden Ratio Symphony! Why so many golden ratios in a relatively simple golden ratio construction with square and circle? Have you seen the attached golden ratio construction before? Three squares (or just two) and circle. For the ratio of segment […]

Are closed simple curves with this property necessarily circles?

Let $\gamma:[0,1]\to \mathbb R^2 $ be a closed simple curve and $\Gamma$ be the region enclosed by $\gamma$. Let $O$ be the center of mass of $\Gamma$. Suppose that any line that goes through $O$ splits $\gamma$ into two regions with equal areas. Is $\gamma$ a circle ? I have no experience in differential geometry […]

5 geometric shapes, all touching each other

I was playing aroud with shapes, which all connected. I managed to get 3 and 4 shapes all connected to each other, but I can’t get 5 to work in 2D. Does anyone have an idea what these shapes are called and also how to get 5 shapes connected? It would be the best, if […]

Integrating the intersection of two cylinders $x^2+y^2=1$ and $y^2+z^2=1$.

Consider two intersecting cylinders. I know the regular way to do this is: $$ \int_{-1}^{1} \left(2\sqrt{1-2x^2}\right)^2dx = \frac{16}{3} $$ This methods integrates the square sides of the solid that you get However, is it just as viable to flip the picture 90° (have the yellow tube going up) and trying to integrate then? You will […]

Bounds on Hausdorff distance via singular values

For some $\delta>0$, let $X$ and $X_\delta$ be two bounded convex polytopes in $\mathbb{R}^n$, defined as $X = \{x \in \mathbb{R}^n : Ax \leq b \}$ and $X_\delta = \{x \in \mathbb{R}^n : Ax \leq b + \delta u\}$ respectively, where $A \in \mathbb{R}^{m\times n}$ (assume $m\geq n$ and $\mathrm{Rank} A = n$), $b \in […]