Articles of geometry

Is there a general formula for creating close approximations of regular polygons on a regular lattice?

Prompted by the question What regular polygons can be constructed on the points of a regular orthogonal grid?: A regular octagon can be approximated on a quad lattice (grid) to about $1\text{%}$ error by knowing that the length of the diagonal of a square is $\sqrt{2}$ (~$1.414$) times as long as its side. With that […]

Normal Intersection of Parabola

If ($ x_1$,$ y_1$) , ($ x_2$,$ y_2$) & ($ x_3$,$ y_3$) be three points on the parabola $y^2= 4ax$ and the normals at these points meet in a point then prove that $\frac{ x_1 – x_2}{y_3} + \frac{ x_2 – x_3}{y_1} + \frac{ x_3 – x_1}{y_2}$=0. Normal Equation: $y=mx-am^3-2am$ Let (x’,y’) be common points […]

Existence of Gergonne point, without Ceva theorem

The intersection at one point (called Gergonne point) of the lines from vertices of a triangle to contact points of the inscribed circle can be proved immediately using Ceva’s theorem. Is there a direct proof that does not pass through Ceva’s formula? Edit: I am hoping for a metric Euclidean proof using lengths and angles […]

How to determine the equation and length of this curve consistently formed by the intersection of Circles

Consider a Point $A$ that moves linearly on the positive $x$-axis with the velocity $1$ m/s and another Point $B$ at a distance $L$ from $A$ with position $(L,0)$. With each forward motion of point $A$ the Point $B$ moves in an arc upward (i.e. along positive $y$-axis) consistently maintaining the distance $L$ from point […]

euler triangle inequality proof without words

today i was studying geometric inequalities and I saw this inequality $$R \ge 2r$$ unfortunately the book did not provided any prove or further explanations. So I just did a little research about it. I find that the name of inequality is euler triangle inequality and there was a simple proof about it If $O$ […]

Simulating simultaneous rotation of an object about a fixed origin given limited resources.

Sorry if the title is a bit cryptic. It’s the best I could come up with. First of all, this question is related to another question I posted here, but that question wasn’t posed correctly and ended up generating responses that may be helpful for some people who stumble upon the question, but don’t address […]

Does there exist a constructible (by unmarked straightedge and compass) angle that cannot be quintsected?

I know that for example an angle of $20^\circ$ cannot be quintsected because an angle of $4^\circ$ cannot be constructed (I’m thinking in terms of (unmarked) straightedge and compass. But an angle of $20^\circ$ cannot be constructed (as above) and I would be interested to see an example of a constructible angle that cannot be […]

Longest pipe that fits around a corner.

This question already has an answer here: Intuitive explanation for formula of maximum length of a pipe moving around a corner? 3 answers

simplify cos 1 degree + cos 3 degree +…+cos 43 degree?

I am currently working on a problem and reduced part of the equations down to $\cos(1^\circ)+\cos(3^\circ)+…..+\cos(39^\circ)+\cos(41^\circ)+\cos(43^\circ)$ How can I calculate this easily using the product-to-sum formula for $\cos(x)+\cos(y)$?

Prove that the point of intersection of the diagonals of the trapezium lies on the line passing through the mid-points of the parallel sides

Prove,by vector method,that the point of intersection of the diagonals of the trapezium lies on the line passing through the mid-points of the parallel sides. My Attempt: Let the trapezium be $OABC$ and that the O is a origin and the position vectors of $A,B,C$ be $\vec{a},\vec{b},\vec{c}$.Then the equation of $OB$ diagonal is $\vec{r}=\vec{0}+\lambda \vec{b}…………….(1)$ […]