Intereting Posts

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well separated points on sphere
Eigenvectors of harmonic series matrix
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Do the Möbius function, totient function, sum of divisors and number of divisors uniquely specify a number?
Why is this formula for $(2m-1)!!$ correct?
How to compute Riemann-Stieltjes / Lebesgue(-Stieltjes) integral?
Show that function $f$ has a continuous extension to $$ iff $f$ is uniformly continuous on $(a,b)$
Maps of discs into surfaces
A set is closed if and only if it contains all its limit points.
Finding the polar cone of the given cone
Three Circles Meeting at One Point
Infinite Inclusion and Exclusion in Probability
If $u\in L^2(U)$ and $u>0$, how to show $u\ln u\in L^2(U)$?
What can be said about the convergence on $|z|=1$?

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.

- Find all integer solutions to $x^2+4=y^3$.
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- Double sum - Miklos Schweitzer 2010
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- Integral $ \int_{-\pi/2}^{\pi/2} \frac{1}{2007^x+1}\cdot \frac{\sin^{2008}x}{\sin^{2008}x+\cos^{2008}x}dx $

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˜‰

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