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!


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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