Bisecting secants in an circumcircular curve
I'm bashing a bit on this problem that has to do with an element of epicycloidal curves. However, this particular element is really about secants.
The Problem:
This is the rough diagram I drew of the problem, as far as I have gotten:
Relevant corrections (but no solutions) will be appreciated
Pathway to the solution:
The following math was contributed by a friend, Dasunt, to solve the red and pink lines.
Radius of Circle = AZ = CZ = BZ
by definition: AC = BC
known side: AB
AD = BD = (1/2 AB)
sqrt( AZ2 - AD2 ) = DZ
CZ - DZ = CD
sqrt( AD2 + CD2 ) = AC = (by definition) CB
Sides AC and BC are now known.
I made the graphics here from base drawings found online - if you want me to I will remove them and put up POV-Rayed examples when I get a spare moment. MLWM 2006
Found Links that I want to remember:
I found this post about science blogging on Science & Politics blog (hopped to from Confessions of a Science Librarian which had a great set of links!)
The Problem:
This is the rough diagram I drew of the problem, as far as I have gotten:
Pathway to the solution:
The following math was contributed by a friend, Dasunt, to solve the red and pink lines.
by definition: AC = BC
known side: AB
AD = BD = (1/2 AB)
sqrt( AZ2 - AD2 ) = DZ
CZ - DZ = CD
sqrt( AD2 + CD2 ) = AC = (by definition) CB
Sides AC and BC are now known.
I made the graphics here from base drawings found online - if you want me to I will remove them and put up POV-Rayed examples when I get a spare moment. MLWM 2006
Found Links that I want to remember:
I found this post about science blogging on Science & Politics blog (hopped to from Confessions of a Science Librarian which had a great set of links!)
posted by RheLynn at 1:34 PM
7 Comments:
Dear RheLynn / Geeky Girl,
I hope you've hopped on the Zeta Beam since you posted your link to see what's shakin' in Ciudad Vleeptron, my vacation planet in the Dwingeloo-2 Galaxy which is entirely governed by Math Nerds on the High Non-Junk Science Council (which explains, amongst other things, why we haven't had a war in 113,222 years).
Nice Proof in Secants, you don't see a lot of Secant Proofs in the blogosphere. Do you use MS_Paint? I find it The Most Excellent Tool for my hand-cranked images, perfect for geometric diagrams, etc.
My post you linked to was illustrated with some filched images of the Salzburg Marionettes performing Mozart's "Abduction from the Seraglio." Well, by odd coincidence, I just published another "Die Entführung aus dem Serail" post -- but this one ain't about the sublime intersection set of Musicke & Mathematics.
Your da Vinci quote ... above the entrance to Plato's Academy in Athens (there was a really great Gyro vendor outside) was writ large:
"Let No One Ignorant of Mathematics Enter."
Another still unanswered PizzaQ:
http://vleeptron.blogspot.com/2006/03/vleeptron-holiday-from-war-get-your_21.html
is easy to Google, but a bit of a challenge if you're Honorable and try to work it out yourself. For this PizzaQ you can also program your PC to get the answer. (In fact I strongly recommend it.)
All math is largely aesthetics and personalities. I'm not an enormous fan of geometry, but I am a Superfreak for Number Theory.
Right now I'm trying to teach myself the hands-on nuts and bolts of the use of the earth's orbital major axis as the baseline (the longest one possible if you're stuck on Earth) of triangles to compute the distances to the stars -- the Parallex Problem. Simple trig, but a sublime application with an astonishingly rich history.
BIG NEWS ABOUT LIEBNIZ! There's a new book that finally nails down the truth about his secret meeting with Spinoza! I'm a Spinoza freak, been to both his homes in the Netherlands. I stalk Spinoza.
Do you have a "take" on Liebniz? I'm a worshipper of Newton and it's possible I've been contaminated with the English prejudice against Liebniz. But Liebniz seems to have been a somewhat creepy and unpleasant fellow.
Again thanks so much for the link! Hop the Zeta Beam to Vleeptron!
Bob
P.S. What's CSS? This question is being asked you by the World's Last QuickBASIC programmer.
QuickBASIC does everything Java and C++ do, only backwards and in high heels.
-- apologies to Ginger and Fred
okay the red-hot new Leibniz and Spinoza book that Oprah's praising to the skies is "The Courtier and the Heretic: Leibniz, Spinoza, and the Fate of God in the Modern World" by Matthew Stewart. This seems to be the first Big Certified Approved book which flat-out states that the two Thinkers actually met up close and personal.
Previously, amongst us Wild & Crazy Spinoza freaks, this meeting was buzzed about sotto voce. The first Creepy Thing about Leibniz in this context is that he wanted to meet Spinoza, he met Spinoza -- but he was too timid or afraid of public controversy to ever admit that the meeting had taken place.
The second creepy thing is a buzz about why Leibniz wanted to meet Spinoza. Nearly everything Spinoza is famous for was published after his relatively early death. But samples and fragrances had been floating around Europe in private letters, and some of it had made it to Leibniz' desk. Knowing that the anathemized Jew was unlikely to publish his own ideas any time soon, Leibniz is accused of having chatted with Spinoza to hoover up some of Spinoza's hotter (but yet unknown) ideas.
If that's not enough, I know more Bad Stuph about Leibniz. Just ask. There are so few people on Earth whom I can tell Bad Stuph about Leibniz to.
Anyway, re
(2^67) - 1
Prime, or Composite? (and what's the Factor?)
I've already writ the skeleton so the proggie can handle integers with hundreds of digits.
Now I actually have to teach the thing how to Add, Subtract, Multiply, Long Divide With Remainder, and take a Square Root (because ordinary Floating Point arithmetic can't handle these Behemoths with the necessary perfect integer precision).
I've written these routines before, and merrily lost that old proggie, so if I'm being a bit slow to climb Everest, it's because I know what a Nine-Aspirin Migraine lies ahead of me. But once I re-invent 4th grade arithmetic and teach it to My Silicon Friend, I am guessing the actual brute-force computation shouldn't take more than an hour.
Okay so like which musical instrument did you play in high school? (Moi, none, my brother got all the musical discipline in the family.)
oh also the proggie has already learned to Add With Carry and so has computed (2^67). By a very nice but slightly unethical coincidence, the proggie then Subtracted 1 and so is standing by with the Required Behemoth.
But now come the gazillion test divisions searching for one which produces Remainder = 0.
Wonderful progress! Oh so much has been going on here -- it isn't even funny. My Descartes Geometry book is in at the library!!! That might kick me off in this direction again -- Leibniz has been lonely sitting on my office shelf buried under tablets of code about paginating user petitions to a live website using only Python... growl... I've also had bronchitis and am coming off the back end of that...which mucked up everything for about a week.
None of that says 'creepy' to me -- weird, obsessed maybe, but creepy is a whole different definition ;o)
band instrument -- they wouldn't let me in. Our school was small and there were only a certain # of students who could get in during seventh grade -- and that was the only year to get in, (you couldn't 'master' the instrument if you started later, so nope, can't try) -- plus it cost $$$$ my mom didn't have so I didn't push on it. My grandfather did teach me to play some piano by ear -- but I gave up on it. I have a fine mandolin in my closet.. but I can only play 'Old Brown's Daughter' on it :o( sad isn't it?
I am hardly one to point fingers about Musical Discipline (I sing and play the theremin -- the instrument where you wave your arms around in the air), but if you can play anything on the mandolin, may I recommend adding these beautiful Parlour Songs from the Past to your repertoire:
"Beautiful Child of Song" (Foster)
"Somebody's Darling"
"Hard Times (Come Again No More)" (Foster)
======
Good grief, didn't know you had a bout with the bronchitis, I'm thrilled that you're beyond it.
I have facsimiles en francais of Descarte's first edition "la Geometrie" (an afterthought bonus appendix tossed into the end of "The Meditations") in which he invents analytic geometry. It's so beautiful, so clear that my crappy essentially non-existant French slices right through it like a hot knife through butter.
I've read a lot about his mathematics, his physics, his philosophy -- but nothing about his Day Job while he was cranking out all this Ethereal Thought in his spare time: as a mercenary officer, a soldier for hire to the highest bidder, king or prince.
The amazingly antithetical things that a single human brain can host, apparently with no internal conflict ...
He never married, but had a daughter he loved profoundly, and her death in childhood apparently caused him unimaginable heartbreak.
Socrates said he did his best thinking while a soldier on guard duty in the snow. Descartes got his inspirations in front of a roaring fireplace. He was a sickly child, and his Jesuit schoolteachers pampered him by letting him sleep late, a habit he retained lifelong.
At the end, down on his luck, he took a job tutoring the teenage Queen Christina of Sweden. He arrived in Stockholm in the dead of winter, and Christina insisted on having her lessons at about 5 a.m. each morning. He was dead within a month. (Later she said she was sorry.)
=======
Anyway Vleeptron wishes to advise you of another math PizzaQ ... shouldn't even make you break a sweat:
http://vleeptron.blogspot.com/2006/06/computation-of-volume-of-fairly-small.html
hey hey hey you want nifty math toys? okay okay the niftiest math toy of all is the POLAR PLANIMETER!!! Check it out!!!
http://persweb.wabash.edu/facstaff/footer/Planimeter/Polar&Linear.htm
(with nifty wiggling Java applet)
okay here's my latest blog post about the Fairly Short Pyramid of Planet Hoon:
http://vleeptron.blogspot.com/2006/06/more-about-fairly-small-flat-top.html
I'm a silly person, I know that.
Hiya Geeky Girl! Hi Husband[Geeky Girl]! Hi Cats[Geeky Girl]!
I hope you check you mail here.
Anyway you might not know that Vleeptron is still archived and accessible, but I had to start up a new blog
VleeptronZ
because blogspot constipated my editing access and I couldn't fix it.
Uhhh ... I wrote a Platonic Object and I'm giving it away for FREE!
http://vleeptronz.blogspot.com/2006/09/free-offer-from-vleeptron-get-free.html
I'm bobmerk@earthlink.net so lemme know if you want it! It's Pretty Spiffy (IIDSSM).
Also check out one of my Earliest Achievements in Science:
http://vleeptronz.blogspot.com/2006/09/more-youth-science-1st-annual.html
It's really icky! Flatworms! Raw liver! Cannibalism!
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