Anti Pythagoras 2: the Line, Ellipse, and Infinite Triplets @drpeyam

Uploaded April 2022 | Updated September 2026, 1 week ago

Anti Pythagoras 2: the Line, the Ellipse, and the Infinity. How to generate infinitely many anti Pythagorean triplets of the form b^2 = a^2 + c^2 - ac The triplets relate the sides of a triangle where one angle is 60 degrees. It is a variation of the Pythagorean theorem from 8th grade math. It has to do with intersecting an ellipse with a line of rational slope and performing some clever algebra like expanding quadratic polynomials and quadratic forms and comparing numerator and denominators of fractions with integers. Then we give an example of a triangle with such a triplet.

Special thanks goes to Ian Fowler who has introduced me to this problem. Ian would like to give a big shout out to Emidio Iacobucci and Frank Cirone for their part in the adventure.

0:00 Introduction
1:40 Rational Slopes
2:40 Finding Nemo
6:00 The triplets are born

Anti Pythagorean theorem: youtu.be/TMR1oYkCKEs

YT channel: youtube.com/drpeyam
TikTok channel: tiktok.com/@drpeyam
Instagram: instagram.com/peyamstagram
Twitter: twitter.com/drpeyam
Teespring merch: teespring.com/stores/dr-peyam

$Anti Pythagoras 2: the Line, Ellipse, and Infinite Triplets Anti Pythagoras 2: the Line, the Ellipse, and the Infinity. How to generate infinitely many anti Pythagorean triplets of the form b^2 = a^2 + c^2 - ac The triplets relate the sides of a triangle where one angle is 60 degrees. It is a variation of the Pythagorean theorem from 8th grade math. It has to do with intersecting an ellipse with a line of rational slope and performing some clever algebra like expanding quadratic polynomials and quadratic forms and comparing numerator and denominators of fractions with integers. Then we give an example of a triangle with such a triplet. Special thanks goes to Ian Fowler who has introduced me to this problem. Ian would like to give a big shout out to Emidio Iacobucci and Frank Cirone for their part in the adventure. 0:00 Introduction 1:40 Rational Slopes 2:40 Finding Nemo 6:00 The triplets are born Anti Pythagorean theorem: https://youtu.be/TMR1oYkCKEs YT channel: https://youtube.com/drpeyam TikTok channel: https://www.tiktok.com/@drpeyam Instagram: https://www.instagram.com/peyamstagram/ Twitter: https://twitter.com/drpeyam Teespring merch: https://teespring.com/stores/dr-peyam$

By popular demand: integral of sin floor 1/x

can you solve this “impossible” trig problem?

life changing integration by parts trick

$Impossible fraction question?$

differential equations model for savings