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

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 […]

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 […]

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?

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

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?

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

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

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 […]

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 […]

Intereting Posts

Why does acceleration = $v\frac{dv}{dx}$
how to determine if two graphs are not isomorphic
Is a function in an ideal? Verification by hand and Macaulay 2
(combinatorics) prove that on average, n-permutations have Hn cycles without mathematical induction.
Matrices – Conditions for $AB+BA=0$
4-ellipse with distance R from four foci
$SL(n)$ is a differentiable manifold
Roots of a polynomial in an integral domain
Why an eigenspace is a linear subspace, if the zero vector is not an eigenvector?
Find intersection of two lines given subtended angle
One-Way Inverse
Help with proof of Jensens inequality
Fourier transform of $\left|\frac{\sin x}{x}\right|$
Why is every answer of $5^k – 2^k$ divisible by 3?
Suppose $Y\subset X$ and $X,Y$ are connected and $A,B$ form separation for $X-Y$ then, prove that $Y\cup A$ and $Y\cup B$ are connected