Articles of geometry

Finding the largest triangle inscribed in the unit circle

Among all triangles inscribed in the unit circle, how can the one with the largest area be found?

Inequality of length of side of triangle

For any triangle with sides a,b,c $$a^2b(a-b)+b^2c(b-c)+c^2a(c-a)\ge 0$$ I tried substituting $a=x+y$, $b=y+z$, $c=z+x$ but well it doesn’t help in any sense except wasting 3 pages that lead to nothing (please don’t mind the joke). Using $a=2R\sin A$, $b=2R\sin B$, $c=2R\sin C$ also didn’t lead to anything for me. Could you give me a hint […]

Decomposition of a nonsquare affine matrix

I have a $2\times 3$ affine matrix $$ M = \pmatrix{a &b &c\\ d &e &f} $$ which transforms a point $(x,y)$ into $x' = a x + by + c, y' = d x + e y + f$ Is there a way to decompose such matrix into shear, rotation, translation,and scale ? I […]

which axiom(s) are behind the Pythagorean Theorem

There are many elementary proofs for the Pythagorean Theorem, but no matter they use areas, similarities, even algebraic proofs, it is not straightforward to tell why it is true tracing back to the (Euclidean geometry) axioms. Are all these proofs equivalent? Do they all track back to the same axioms?

Sum of Angles in a Triangle.

Can anyone please explain how to form a better idea in understanding sum of measures of angles in a triangle is $180^\circ$ ?

How to find $n+1$ equidistant vectors on an $n$-sphere?

Following this question, Proving the existence of a set of vectors, I’m looking for a way to find $n+1$ equidistant vectors on a Euclidean $n$-sphere. For $n=2$, you can pick the vertices of any equilateral triangle. For $n=3$, pick a tetrahedron. What about larger dimensions?

How come $32.5 = 31.5$?

Below is a visual proof (!) that $32.5 = 31.5$. How could that be?

Determinant of transpose?

$$\det(A^T) = \det(A)$$ Using the geometric definition of the determinant as the area spanned by the columns could someone give a geometric interpretation of the property? Thanks!

Geometric construction of hyperbolic trigonometric functions

If we have a circle we can geometrically construct the trigonometric functions as shown. The functions all derive from sin and cos. If we say that the circle is a conic section and imagine it on the cone we can draw a hyperbola perpendicular to it. I believe that the hyperbolic trigonometric functions can be […]

Is a sphere a closed set?

The unit sphere in $\mathbb{R}^3$ is $\{(x,y,z) : x^2 + y^2 + z^2 = 1 \}$. I always hear people say that this is closed and that it has no boundary. But isn’t every point on the sphere a boundary point? Since for every point $x$ on the sphere, any open ball in $\mathbb{R}^3$ (defined […]