Pascal's theorem | Wikipedia audio article
Channel: wikipedia tts
Date: 30/12/2018

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Pascal%27s_theorem


00:00:35 1 Euclidean variants
00:01:31 2 Related results
00:03:55 3 Hexagrammum Mysticum
00:04:57 4 Proofs
00:06:06 5 Proof using cubic curves
00:06:18 6 Proof using Bézout's theorem
00:07:51 7 A Property of Pascal's Hexagon
00:09:37 8 Degenerations of Pascals's theorem
00:10:16 9 See also
00:10:41 10 Notes
00:11:05 11 References
00:11:16 12 External links
00:11:25 A Property of Pascal's Hexagon
00:12:54 Degenerations of Pascals's theorem
00:13:33 See also



Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago.

Learning by listening is a great way to:
- increases imagination and understanding
- improves your listening skills
- improves your own spoken accent
- learn while on the move
- reduce eye strain

Now learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone.

Listen on Google Assistant through Extra Audio:
https://assistant.google.com/services/invoke/uid/0000001a130b3f91
Other Wikipedia audio articles at:
https://www.youtube.com/results?search_query=wikipedia+tts
Upload your own Wikipedia articles through:
https://github.com/nodef/wikipedia-tts
Speaking Rate: 0.9960104251139974
Voice name: en-US-Wavenet-D


"I cannot teach anybody anything, I can only make them think."
- Socrates


SUMMARY
=======
In projective geometry, Pascal's theorem (also known as the hexagrammum mysticum theorem) states that if six arbitrary points are chosen on a conic (which may be an ellipse, parabola or hyperbola in an appropriate affine plane) and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon (extended if necessary) meet at three points which lie on a straight line, called the Pascal line of the hexagon.
The theorem is also valid in the Euclidean plane, but the statement needs to be adjusted to deal with the special cases when opposite sides are parallel.