Intereting Posts

Help me solve this olympiad challenge?
Gradients of marginal likelihood of Gaussian Process with squared exponential covariance, for learning hyper-parameters
Modified gambler's ruin problem: quit when going bankruptcy or losing $k$ dollars in all
Proving Gaussian Integers are countable
Show that 3 is the most efficient number.
Spectral Measures: Spectral Spaces (II)
Why is it called a series?
Infinite Prime Numbers: With Fermat Numbers
Help with the Neumann Problem
Can this intuition give a proof that an isometry $f:X \to X$ is surjective for compact metric space $X$?
Chess Piece Combinations
Locally Constant Functions on Connected Spaces are Constant
Compute variance of logistic distribution
Proof that $\Bbb Z$ has no other subring than itself
Square root confusion?

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 but this should not limit the answers. If you can prove it using algebraic K theory, go ahead.

- Solve $\lim_{x\to +\infty}\frac{x^x}{(\lfloor x \rfloor)^{\lfloor x \rfloor }}$
- No function $f:\mathbb{Z}\rightarrow \{1, 2, 3\}$ satisfying $f(x)\ne f(y)$ for all integers $x,y$ and $|x-y|\in\{2, 3, 5\}$
- Roots of unity filter, identity involving $\sum_{k \ge 0} \binom{n}{3k}$
- combinatorics olympiad problem - sort out a schedule
- Combinatorial Proof Of A Number Theory Theorem--Confusion
- How to prove that exists distinct $x_1,x_2 \in(a,b)$ such that $f '(x_1)f '(x_2)=1$?

- IMO 1988, problem 6
- How to prove $\cos \frac{2\pi }{5}=\frac{-1+\sqrt{5}}{4}$?
- Irrational painting device
- Proving $AE+AP=PD$ In a Certain Right Triangle
- A triangle with vertices on the sides of a square, with one at a midpoint, cannot be equilateral
- Minimum period of function such that $f\left(x+\frac{13}{42}\right)+f(x)=f\left(x+\frac{1}{6}\right)+f\left(x+\frac{1}{7}\right) $
- block matrices problem
- How to prove there exists $n_{1}a_{n_{0}}+n_{2}a_{n_{1}}+\cdots+n_{k}a_{n_{k-1}}<3(a_{1}+a_{2}+\cdots+a_{N})$
- Find all positive integers $n$ such that $n+2008$ divides $n^2 + 2008$ and $n+2009$ divides $n^2 + 2009$
- Intersection of two parabolae

It is a conclusion of a degenerated case of the Brianchon’s theorem:

For a hexagon $ABCDEF$ with an inscribed conic section the lines $AD$, $BE$, $CF$ coincide in a common point.

Just set $A$, $C$ and $E$ to be vertices of the triangle, and $B$, $D$ and $F$ the points where the incircle touches the triangle. Some immediate conclusion: the Gergonne point could be generalized to ellipse and other conics. Quick search reveals this idea is not new and there is even some literature on it, e.g. this nice pdf and also this page.

I hope this is what you were looking for ðŸ˜‰

- The sequence of improper integrals of the form $\int\frac{dx}{1+x^{2n}}$
- Is the Michael line star countable?
- Does the notion of “rotation” depend on a choice of metric?
- How do we check if a polynomial is a perfect square?
- Totally disconnected space
- Circular logic in set definition – Tautology?
- On the greatest norm element of weakly compact set
- Number of distinct numbers picked after $k$ rounds of picking numbers with repetition from $$
- $\epsilon$-$\delta$ proof, $\lim\limits_{x \to a}$ $\frac{1}{x}$ = $\frac{1}{a}.$
- Stationary distribution for directed graph
- Find the pdf of $\prod_{i=1}^n X_i$, where $X_is$ are independent uniform random variables.
- Matrix with non-negative eigenvalues (and additional assumption)
- Geodesics (2): Is the real projective plane intended to make shortest paths unique?
- How to find $\sum_{i=1}^n\left\lfloor i\sqrt{2}\right\rfloor$ A001951 A Beatty sequence: a(n) = floor(n*sqrt(2)).
- Using Integration By Parts results in 0 = 1