A Challenging Integral

Some weeks back, during a Math Teachers Workshop being conducted by the AMTI at the Institute of Mathematical Sciences (IMSc), Chennai, for a group of teachers from the Jawahar Navodaya Vidyalas, one of the teachers (Mr Sasi Kumar, of Palakkad, Kerala) posed a challenge to the group: the evaluation of a definite integral. It proved to be very challenging indeed. Indeed, no one got it.

On getting back and struggling with it for some days I finally cracked it. I have posted the solution here: ToughIntegral.

What is interesting to me is that it resisted all my attempts at an elementary solution. I wonder if there exists any such solution at all. My solution ultimately had to depend on Cauchy's Residue Theorem. If some reader finds an elementary solution, I would be most interested in seeing it.

Morley's Miracle

This GeoGebra applet demonstrates the famous theorem discovered by Frank Morley towards the end of the nineteenth century.

Triangle ABC is arbitrary. Each internal angle of the triangle is trisected, and the trisectors meet at points P, Q and R, as shown (the trisectors closest to BC meet at P, the trisectors closest to CA meet at Q, and the trisectors closest to AB meet at R).

And now the miracle: triangle PQR is equilateral, regardless of the shape of triangle ABC!


Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)


You can test it out by dragging the vertices using the mouse!

For more on Frank Morley, you can study the Wikipedia site MorleyWiki: or this site: MorleyBiography. I was impressed by the fact that he was a very good chess player and even managed to beat the great Emanuel Lasker while Lasker was the World Chess Champion.

NASA Finds Earth's First Trojan Asteroid

A Trojan asteroid has been found by NASA in the same orbit as the Earth! Here is a link that tells you about it:

NASA's WISE Finds Earth's First Trojan Asteroid - NASA Jet Propulsion Laboratory

And here is an animation that depicts the orbit of the asteroid. It is quite striking that the orbit takes it regularly above and below the plane of the Earth's orbit; enjoy! --

Euler, Madhava and pi

Euler and the Basel problem -

A pattern of signs

Signs