Articles of geometry

Relating area to a line intersecting with a point.

I really could use a hint with this following problem: If a line L separates a parallelogram into two regions of equal areas, then L contains the point of intersection of the diagonals of the parallelogram. The figure shows a line L horizontally through the sides of the parallelogram. This creates two trapezoids and I […]

Circle-Cirle Intersection

Sorry for asking a question like this (can’t comment as my reputation is too low). Could someone please explain this answer to me? Particularly the part, where the orthogonal vectors are declared? Why are the vectors declared as such? What is the reasoning or math behind declaring them in such a way? Given the points […]

Geometrical construction by Descartes/ Oblique hyperbola

I wish to share verification/completion of the original construction of the oblique hyperbola by Descartes. At first I was looking at the construction of right branch only and temporarily concluded that the sketch was in error as it was not included in original. But then realized that the incomplete sketch defined one branch only. And […]

Finding the no. of possible right angled triangle.

The inradius of the triangle is 2013 of a right angled triangle with integer sides. So how many right angled triangles can we form with this specification. Thanks in anticipation.

Chain of Circles

A chain of four circles centered at A, B, C, and D are touched on one side by the line GH and on the other side by a circular arc EF centered at O. Find the area of D in terms of the areas of A, B, and C The hint is to find the […]

Prove sum of distance from triangle vertices to a point inside triangle is more than semiperimeter and less than perimeter

If $O$ is a point inside $\triangle ABC$,Prove: $$\frac{\overline{AB}+\overline{BC}+\overline{CA}}{2}<\overline{AO}+\overline{BO}+\overline{CO}<\overline{AB}+\overline{BC}+\overline{CA}$$ Figure Things I have done: I was able to proof second part $$Proof \space of \space second \space part\\\\ \begin{array}{ l| l } \hline Statement & Reasoning \\ \hline \overline{AB}+\overline{AE} > \overline{BE} & Triangle\space inequality \\ \overline{OE}+\overline{CE} > \overline{OC} & Triangle\space inequality \\ \overline{AB}+\overline{AC} > \overline{OC} […]

Points on a plane and triangle

Given any three non-collinear points on a plane. Can there be an equilateral triangle such that the points lie on the perimeter of the equilateral triangle ?

How to calculate the radius of a circle inscribed in a regular hexagon?

If I know how long one side of a regular hexagon is, what’s the formula to calculate the radius of a circle inscribed inside it? Illustration:

How to determine 2D coordinates of points given only pairwise distances

I have a set of 2D points in which each pair has a known Euclidean distance between them. How can I go about determining an arrangement of them? I understand there is not a unique solution in general, but for the sake of my question, assume one point is fixed at the origin. Mathematically, we […]

How to generalize parallelograms to non-Euclidean spaces?

Question: How can one generalize parallelograms to non-Euclidean spaces? In particular, how can one generalize parallelograms to Finsler manifolds which are not necessarily affine spaces (i.e. to “spaces with norms which don’t satisfy the parallelogram law”)? Or at least to Riemannian manifolds which are not affine spaces? The dream of course would be a generalization […]