They have yet to respond.
Tuesday, May 6, 2025
The Science of Victory - an underrated war manual
It was written by General Suvorov, one of history's few undefeated commanders. He even defeated Napoleon.
He summarizes winning as: eyes, speed, impact. That means find the enemy before he finds you, move in quickly, and hit him as hard as possible at his weakest point.
Pobeda means victory in Russian, and there is a verb for which means "to be victorious". It is the second word of his book. The Wikipedia article on Suvorov is excellent.
Tuesdays With Derpy - A warning from the future
The old man awoke. The lights were on, and he was still wearing the same clothes he had on 12 hours ago. “Christ, did I drink until I passed out *again*?”, he wondered to himself. He struggled out of bed and turned on the holoprojector, already set for NPR. Instantly, a decrepit, 104 year-old Ken Burns was in mid-sentence: “…destroying all hitherto known norms and…[wheeze, hack, cough] of our democracy. But this was only the beginning. A female voice-over spoke: tune in next week for part 68 of Ken Burns’ seminal series: Surviving the Horror of Trump’s America. If you would like to contribute to Ken Burns’ anti-gerisol fund, please uptwinkle and wait for the prompt to scan your Patriot bar code.
Her vocal fry and uptalk made the old man wince.
Well, time to make breakfast, the old man thought. He put the soylent hotdogs in the sink to thaw. A passing maglev train rattled his room in the cage motel. Transients welcome, said the friendly neon sign on its facade. “How often does the train go by?”, he remembered asking his roommate Jasper 12 years ago. “So often you won’t even notice.” Jasper was dead now. He got his beard caught in a woodchipper.
Golden years, shmolden years, he scoffed to himself. I’m running on fumes with 4 flat tires.
Ah, but today was Tuesday, and that was his favorite day because the young man from college would come to visit and talk. He promised to write a book about their chats.
“How about we call it ‘Tuesdays with Derpy’ ?”, suggested the old man.
***
The old man left his room in the cage motel and walked downstairs. He preferred the stairs even though it made his legs ache because the stairwell was quiet. Almost no one used stairs anymore, so it was one of the few places in the urban world that wasn't covered with ads and filled with noise.
If rats bought soylent, there'd be ads in the sewers, the old man thought to himself.
Out in the street, he made his way to the park. He was going to meet the college kid there to feed tofu to the ducks. He reminded himself to tell the college kid that people used to feed *bread* to ducks, but that was before the War on Gluten.
War on this, war on that. So many of them. He couldn't remember a single time in his life when there hadn't been a war.
Across the street, a group of protesters stood in front of a pharmacy. From their slogans and signs, he could tell they were Bacteria Rights protesters.
"Hey-hey! Ho-ho! Flu vaccines have got to go!", they shouted in rhythm as they uptwinkled and danced.
The old man crossed the street to check things out. He overheard a man in a white lab coat trying to reason with a protester.
"Look, first of all, the flu is a virus, not bacteria. Please listen. I'm a pharmacist."
The scrawny, pink-mohawked, nose-ringed protester was silent for a moment as her(?) face twisted into a grimace.
"Don't man-splain to me, racist. Vaccines have *electrolytes*! The science is settled!"
"Do you even know what an electrolyte is?!" The pharmacist's faced flushed and his voice shook.
The old man decided it was time to take one for the team. He cupped his hands to his mouth and shouted in rhythm:
Slogan, slogan, buzzword, rhyme! We are here to waste your time! We got bullshit, yes we do! We got bullshit how 'bout you?
There were a few stifled laughs from the protesters and a stony look of hate from Pink Mohawk. She(?) blew a whistle and shrieked: "Haaaate...speeeeeeeeech!" It was like the screech of a red-tailed hawk with a women's studies degree.
Flashing blue lights and tentacle-mounted cameras popped out of the sidewalk. Sirens blared. A robotic voice announced: Citizens, identify the thought criminal!
The protesters screamed and pointed at the old man. A camera tentacle came 6 inches away from the old man's face. It spoke to him: Citizen, explain yourself.
The old man answered: Lo siento, no hablo ingles. Soy solamente un hombre viejo cansado. Que es la problema?
The camera tentacle withdrew and fixed its gaze on the protesters. It spoke to them: you are all in violation of section 62 of the Verbal Morality Statute: 1st degree othering a non-English speaker. You are hereby sentenced 10 hours of sensitivity re-education and your social media accounts will be suspended for 1 hour. You have 20 seconds to comply. 19, 18, 17..."
The protesters kneeled and put their hands behind their backs. The camera tentacles roped them together and a security bot led them away like a very sad, patchouli-scented train.
The pharmacist thanked the old man and shook his hand. The pharmacist also said that if the old man ever needed certain things [cough] anti-gerisol [cough], come to his pharmacy.
The old man walked on. He passed Veterans of Antifa Post 1138. The wall had a mural of 6 Antifas throwing milkshakes at a guy wearing a MAGA hat.
Christ, what a bunch of assholes he thought. He decided to save that fight for another time.
It was late in the afternoon when he finally reached the park and found the college kid.
What took you so...you must have gotten in another fight, the college kid answered his own question.
Just gimme some tofu, said the old man as he put out his hand.
As he tossed a squishy cube into the pond, he mentioned that when he was a kid, people fed *bread* to ducks.
Oh really? That must have been before the War on Gluten, said the college kid.
The old man smiled. He was happy he remembered.
***
The old man and the college kid slouched on the couch and drank bottles of Tolerance Lager. The holoprojector was playing NPR news.
The announcer was breathless: this morning's mass shooting was the worst in history. With us a now is a spokesman from the National Ray Gun Association.
My first question sir, is why does anybody need a plasma rifle in the 40-watt range?
The holoprojector displayed the rifle, with its gleaming chrome body and graceful curves. It was an elegant weapon from a more civilized era.
I'd buy that for a dollar, said the old man.
Yeah, me too said the college kid.
I think a better question, said the spokesman, is why are you making this about the weapon rather than the shooter?
Because your organization is the reason these murder machines are available, screeched the announcer.
Ooh, Madame Vocal Fry is getting that righteous anger, said the old man.
I think I know why they haven't said anything about the shooter, said the college kid as he scrolled his Eye Phone for more news.
Oh? Who did it, asked the old man.
It appears that the shooter was a member of a certain community...the college kid trailed off.
Xenomorphs? Yeah, that's been happening a lot lately, said the old man.
The college kid gave him the stink eye and nudged him in the ribs.
Oh, right. The old man counted down from 3 with his fingers, then they both got up and shouted in unison:
We totally respect the Xenomorph community. Xenomorphophobia is hate speech!
There, that should keep the damn Thought Police happy, muttered the old man.
Our next guest a is a spokesman from the Council of American-Xenomorph relations, said the announcer.
Special thanks to Amazon, who built a special purpose Alexa to translate, the announcer added.
It was a lobster-like creature. For a few moments, it growled, buzzed, and hissed as its antennae swayed, and its mandibles fidgeted.
Great to be here, said the bubbly female voice emanating from the Alexa.
Sir, remind us again about the purpose of your organization, asked the announcer.
More growls, hisses, and buzzes. Then the voice of Alexa again.
The Council of American-Xenomorph relations works to build respect and understanding among all the sentient life forms of this planet.
It is part of the larger effort to build the Greater Galactic Co-Prosperity Sector. I am proud to represent it.
It fills my abdominal and thoracic hearts with joy.
Ah, how nice, said the announcer. Forgive me, but I must ask if you have a comment on today's tragedy, asked the announcer.
Even more growls, hisses, and buzzes. It sounded like someone threw a beehive in a fire and then doused it with water.
Of course, it is terrible, and it saddens us greatly that the shooter was one of our own.
But what about the National Ray Gun Association and white males? Surely they share some of the blame. We must be careful not to demonize an entire community, especially one as committed to peace and unity as ours.
Wise words. Thank you so much for being here, said the announcer.
My pleasure, the creature gargled in nearly intelligible English.
And now we turn to world news, specifically the ongoing turmoil in Nee-kah-rrrawg-wah, said the host.
Ugh, enough already, groaned the old man as he turned off the holoprojector.
Let's go feed tofu to the ducks, said the college kid.
Great idea. At least the ducks won't give me some half-baked sermon said the old man.
***
You got a letter, and it looks important, said the college kid as he entered the old man's room.
Ah, the good ol' US Postal Service, thought the old man. Those guys are giving the cockroaches a run for their money.
It was a angry-looking letter with lots of red ink and the words FINAL NOTICE- IMMEDIATE ACTION REQUIRED!
I forgot where I put my reading glasses, said the old man. Could you read it to me please?
The college kid opened it and began reading: You are receiving this notice because you are delinquent in your PanopticonBook account.
To avoid fines and penalties, you must log into your account at once and post the following:
1. a comment praising your local government
2. a picture of the last meal you ate
3. a comment praising the state government
4. a video of you criticizing something; see below for the list of acceptable targets
5. a comment praising the national government
6. a comment giving your full name, age, address, employer, Social Security number, and phone number
7. a picture of your face no more than 24 hours older than the postmark of this notice
8. a lolcat picture
The old man sighed heavily. It's bad enough I get bombarded with their propaganda and now they're forcing me to write it for them, he said.
Do you even have a PanopticonBook account, asked the college kid.
I did a really long time ago, said the old man. It was called something else back then. That was before the gubmint seized it.
How did things get like this, asked the college kid.
The old man paused to gather his thoughts. He rubbed his chin. Then he spoke:
About 200 years ago, life was tough for everyone. Most people had to work very hard just to stay alive and many died young.
The world was also a much more dangerous place. And even rich people couldn't buy the things we take for granted today.
Slowly, over the next hundred years or so, technology improved, and things steadily got better.
I'd say overall, technology peaked when I was a kid. The last great invention was the internet and that was about 70 years ago.
Anyway, as things got better, life got easier, and people became dumber and weaker.
In the past, you had to be some combo of smart, strong, sexy, and lucky to live long and prosper.
But as technology made everyone richer, we reached a point where any moron or jackass could survive and have kids.
That process was nearly complete when I was about your age. Things took a turn for the worse when you-know-who got elected.
When you-know-who got re-elected, the country collapsed. The red countryside became deep red and the blue cities became deep blue.
It's been that way for about 40 years now and it will probably stay that way, the old man said as he plopped down on the couch.
Cheer up, said the college kid. I got you a present.
It was a belt. The buckle was in the shape of an onion.
You know, I didn't really wear an onion on my belt, said the old man. That was just a joke from an old TV show.
...T...V?, asked the college kid. Is that like NPR?
The old man laughed. Yeah, I guess it was. I just wasn't smart enough to see it back then, said the old man.
Well, I guess I should go take some of those whatcha-call-it...selfies?, the old man wondered aloud.
The college kid laughed. Heh, I learned about those in history class. What a stupid fad!
***
The hovercar trundled down the dusty path.
Keep your eyes peeled for the sign, said the old man in the passenger seat.
What's it say again, asked the college kid.
Whizzing Cheese Republican Reservation, said the old man.
The last of the Republicans, he added.
Well, we got another hour at least on the road. Any pearls of wisdom to share, asked the college kid.
The old man leaned back and rubbed his chin thoughtfully. After a few moments, he spoke:
There are basically 2 types of people in the world. Nobody is all the way one or the other, but everybody is closer to one side.
The first kind, I'll call him the Hank Hill type. These guys are great: honest, polite, hard-working, careful.
They are the reason toliets flush, lights turn on, and planes don't fall out of the sky. Without them, we'd all be living in caves.
The other kind, I'll call him the Keith Richards type. These guys are great too: wild, creative, fun, passionate.
Every song, book, or movie you've ever liked, it came from the Keith Richards tribe. And a lot of great, new ideas too.
So we need both, but there has to be a balance. When too many Hank Hill types are in charge, life can get really boring.
And when too many Keith Richards types are in charge, everything goes to hell so fast it makes a dumpster fire look like Swan Lake.
Somewhere along the line, too many Keith Richards types got in charge. And not the smart ones either. Things started to get bad.
The Hank Hill types knew something was wrong and they tried to fix things. They went about it the only way they knew how.
They wrote long-winded letters to the editor. They went to protests dressed up like Ben Franklin. They went to school board meetings.
While they were doing that, the Keith Richards types were making songs, movies, TV shows, and college classes for their side.
Somewhere along the line, the Hank Hill types forgot how to tell a good story, and that is why they lost.
It's like what I read in an old book: Every trail has its end, and every calamity brings its lesson!
***
The old man and the college kid slouched on the couch and drank Equality pale ale. They were watching NPR, because that was all there was.
The announcer spoke: And now, we proudly present a new documentary by Ken Burns- Heroes of Antifa.
Beautiful, dramatic music played over a slow-motion, sepia montage: 2 Antifa guys pushing over a trashcan, a scrawny Antifa throwing an awkward punch,
a mob of Antifa struggling to pull down a statue, Antifa knocking MAGA hats off eldery people...
Gee, I sure hope they don't break their arms from patting themselves on the back too hard, said the old man.
The young man's face contorted as he struggled to understand. Then he said: they...they really thought they were amazing heroes for *that*?!
Oh, you don't know the half of it, said the old man. Back in the day, seemed like everyone of them thought they were storming the beaches of Normandy for
retweeting "orange man bad" stuff. You know where the name Antifa comes from, don't you?
No, said the college kid.
It stands for anti-fascist, you know, like Nazis and such, said the old man.
I thought Nazi and fascist were just insults, like racist or moron or Republican, said the college kid.
The old man buried his face in his hands and sighed. No, all those are real things and they are not the same, said the old man.
So what's the difference, asked the college kid.
The old man paused and said: I could spend hours explaining the differences, but the gist is Nazi, fascist, and racist refer to evil and false beliefs, moron means
not smart, and about half the people in this country are Republicans of some sort, even if they are reluctant to say so.
So how did those words turn into insults, asked the college kid.
Social media, said the old man. Our prog overlords had a monopoly on that, and anyone who said something they didn't like was labeled racist, Nazi, etc.
That's really stupid and sad, said the college kid.
Yeah, said the old man. Over time, all the dissenting voices were silenced. People became afraid to speak their minds, and so the vicious cycle began.
I've learned a lot since I met you, said the college kid. I should go home and work on our book. Have a nice night.
Half an hour later, the old man was relaxing on the roof of the cage motel. He took lazy puffs on a cigar and looked at the sky.
Ortega. Palmer. Kowalski. He didn't know what hurt more- that they were gone or that he had trouble remembering their faces and names.
He crushed out the cigar and went to bed.
***
It was a dreary day. The Old Man wanted to liven things up. What kind of snitching goes on these days, he asked the college kid.
Oh, there are so many, said the college kid: the Department of Hurt Feelings, The Preferred Pronoun Police, the Bureau of Microaggressions...
OK, I get the idea, said the Old Man. Say, pick one of those places and put them on speaker phone with me.
You got it, said the college kid. The sound of a ring tone filled the room washed over the room like waves on a beached whale for a few moments.
Department of Emergency Re-education, state your denunciation, said the robo-voice.
The Old Man adopted a falsetto: yes, this is Sue DeNim of 123 Fake Street. I wish to report a misgendering I witnessed at the Veterans of Antifa Post 1138.
A Bias Response Team has been dispatched, shouted the robo-voice before the call abruptly ended.
Call another, said the Old Man. The college kid scrolled his EyePhone for a moment. More ringtones.
Hate Speech Hotline, please describe the thought crime, chimed a robo-voice.
The Old Man did a bad Colombo impression: My name is Hugh Jass, and ah, I live close to the, ah, Antifa Vet post...
The robo-voice spoke again: stay on the line, citizen, this is a serious matter and it is a felony to make an incomplete report.
It sounded like a talking smoke alarm that just found out the airline lost its luggage.
The Old Man stayed in character: well, lemme just ask ya one thing- is it a crime when some Antifa guys don't use the right pronouns?
A sympathy squad has been dispatched with extra crayons and puppies, screeched the robo-voice.
Call one more, said the Old Man with satisfaction. Even the ringtones started to sound like stifled laughter.
Misogny and Mainsplaining Ministry, chirped the robo-voice. Please describe the nature of the sexism.
The Old Man did a bad John Wayne impression: Well, pilgrim, my Name is Mike Hunt, and if there's one thing that gets me angry, it's sexism.
Go on, said the robo-voice, like a microwave announcing a freshly cooked bag of popcorn.
The Old Man continued: well, ah...the thing is, the sexists are the guys from the Department of Re-Education. One of them said...something so awful...
Go on, pleaded the robo-voice. There was a faint hint of an old dial-up modem in its speech.
The Old Man continued with feigned angst: xe said...xe said...there are only 2 genders! Oh, it was just the most problematic thing!
Gender Studies commandos are on the way, shrieked the robo-voice.
The Old Man breathed in slowly and sat in his recliner. And now we sit back and watch the fireworks. Put on NPR, he said.
The announcer was somber: Tragedy struck today at Antifa post 1138 as multiple goodthink enforcement agencies rushed to the scene of a badthink hatecrime sadcry.
In the ensuing chaos, the puppies escaped their expert handlers, ate most of the crayons, and had accidents all over the post's rec room.
The holoprojector displayed a court-room style color pencil drawing of twig-armed cretins screeching at each other while puppies shat everywhere.
We reached out to a Jedi Moonshine Lumberg, one of the veteran survivors of today's incident for comment, said the announcer.
A bald old man with a nose ring and a haggard face spoke haltingly: today was terrible. It was almost as bad as the time I sprained my wrist from throwing a milkshake.
***
What's this?, said the college kid as he held up some framed ribbons.
Oh, that's just a few souvenirs from some stuff I did a long time ago, said the old man.
Liar, said the college kid.
Ok, fine, I tried to be modest, said the Old Man. Get us some beers, will ya?
The Old Man took out his dentures and used them as a bottle opener. The cap of the Progress Pilsener clattered to the floor.
He began: I re-upped after my first few years in the Army and got press-ganged into the Space Force. They sent off to some god forsaken rock. I think it was called Chalupa Prime. It's one of the moons orbiting the gas giant Quilmes.
You were in the War of the Argentine Moons?, asked the college kid.
Well, I was *there*, said the Old Man, but I didn't see much combat. You see, I was already an old man by military standards, so it was mostly rear echelon stuff. Some of it was interesting, but on the whole, kind of a yawn fest. Except for the Crustaceanoids, of course. We called them Shellbacks. We were there to protect the refinery from them. You see, at the time we required more Vespene gas. And so began Operation Crab Cake. Of course, there was plenty of down time, enough for us to mingle with the locals. There were these things. They best way I can describe them is lobstergirls. Well, they were always ready for a good time. I heard they gave good clawjobs, but I was too scared to get one. Anyway, where was I?
Sex with space alien lobster girls, said the college kid.
Oh yeah, that. Good times, lemme tell ya. Old Bay is their aphrodisiac; remember that if you're ever on Chalupa Prime. That was a long time ago. I retired from the military a few years before it got renamed the Social Justice Defense Forces. And that's pretty much it.
The Old Man chugged the rest of his beer and put the bottle on the end table.
***
The voice on the holoprejector spoke: and now, NPR proudly presents part 89 of the award-winning documentary by Ken Burns: Surviving the Horror of Trump's America.
The aforementioned 104-year old film maker spoke haltingly: In the spring of 2020, a devastating plague swept the nation. Historians agree that the blame lies largely with Trump, who ignored and gutted the very agencies charged with protecting against such disasters. Instead of coordinating with the responsible Chinese, who had been coordinating with the World Health Organization to fight the pandemic from day 1, Trump chose instead to engage in wild conspiracy theories which forced the nation's sensible Democrat governors to issue lock down orders...
The old man was so angry, he spit out the beer in his mouth and threw the mostly full can at the screen. That's not what happened, you goddamn worthless candy-ass chickenshit liar!
The monitoring unit on the wall buzzed and informed the Old Man he was fined 10 credits for violation of the Verbal Morality Statute.
Thursday, April 24, 2025
The Groogle's Doodle and the Strudel Bugle
***
I put Lewis Carroll, Dr. Seuss, and Shel Silverstein into my mental blender, and this is what came out.
***
A groogle named Dougal made a doodle of a noodle-eating poodle
"Oh, what fine work I've done!" said the groogle
the groogle blew a bugle, the signal for strudel
but even after the groogle blew the strudel bugle
the groogle's servant came without a strudel
"Apologies, my lord Dougal", said the abashed servant
who had to muddle through a puddle on the double
"We're out of strudels and I lack the recipe"
At this, the groogle (named Dougal) used Google to find strudel
"Ah ha!" said the droogle named Dougal, "let's take the flagon wagon"
and so the droogle and his servant named Foozle set out for strudel
the flagon wagon stout and sure, for it was a gift from a dragon
thus the droogle named Dougal and his servant named Foozle set out
in the dragon flagon wagon in search of strudel found on Google
"When you see the strudel", said the groogle named Dougal
to his servant named Foozle, "be sure to blow the strudel bugle"
Blowing the strudel bugle was the only way to scare off the
noodle-eating poodle that the groogle named Dougal had doodled
Their journey was not easy; they had to muddle through a puddle
in the dragon flagon wagon; there was a bungle in the jungle
when a killer gorilla flotilla from Manilla attacked the groogle
named Dougal and his servant named Foozle just as Foozle spotted
the strudel and blew the strudel bugle from the dragon flagon wagon
a killer gorilla from the Manilla flotilla grabbed a paddle for the battle
that was a jungle bungle and the groogle and Foozle tried to muddle through a puddle
but the puddle paddle battle with the killer gorilla flotilla from Manilla for
the dragon flagon wagon began just as Foozle blew the strudel bugle
It was a raucous melee when entered the fray none other than the noodle-eating poodle
whom the droogle named Dougal had doodled before he Googled for strudel!
For you see, when Foozle blew the strudel bugle, he summoned the noodle-eating poodle
named Kitten Kaboodle to the midst of the dragon flagon wagon puddle paddle battle
with the killer gorilla flotilla from Manilla
Kitten Kaboodle seized and blew the strudel bugle which signaled all the noodle poodles
to converge on the dragon flagon wagon of the droogle named Dougal and his servant named Foozle
The killer gorilla flotilla from Manilla was about to flee when the poodle named Kitten Kaboodle
offered strudel to them, for that is the custom whenever a noodle poodle hears the call of the strudel bugle
In thanks for the strudel, the droogle named Dougal passed the strudel bugle to the noodle poodle
named Kitten Kaboodle and ended the dragon flagon wagon puddle paddle battle with the killer gorilla flotilla from Manilla
Once back at home, the groogle named Dougal made another doodle of the noodle poodle named Kitten Kaboodle who now was the keeper of the strudel bugle
Sunday, April 20, 2025
perfect squares and the factorization of semiprimes
I'll start with a few simple examples.
11*13 = 143
12*12 = 144
13 - 12 = 1
12 - 11 = 1
144 - 143 = 1
1 = 1^2
Hmm. Interesting.
7*11 = 77
9*9 = 81
11 - 9 = 2
9 - 7 = 2
81 - 77 = 4
4 = 2^2
So if a and b are prime, and c is an integer halfway between them, then:
sqrt(c^2 - a*b) = a - c = c - b
So given any semiprime, if we round it up to the nearest perfect square and find the difference between that and the semiprime, we can quickly factorize. The integer factorization problem is currently believed to be too hard to solve with current methods. It is also the basis for RSA encryption.
I like to visualize semiprimes as rectangles made up of unit squares. If you take one of the rows/columns from the short side and stick it onto the long side, you'll see a gap of how many unit squares must be added to make the whole thing a perfect square. That gap will be the shape of a square.
Semiprimes can be thought of as perfect squares with a missing square-shaped piece of one corner.
OK, time for a bigger example.
991*997 = 988027
Assume we want to factorize 988027. If we take the square root of that, we get almost 994, which is halfway between 991 and 997. If we square 994 and subtract it from the semiprime, we get a difference of 9 which is a perfect square. This proves that 994 is halfway between the factors of the semiprime and that the difference between each factor and 994 is 3 (square root of 9).
Is this end of the integer factorization problem? Maybe. Rigorous proofs are not my strength.
Good Storytelling vs. Good Writing
These two lists might explain the difference:
only sold well (over a million copies)
Alice in Wonderland, Dune, Pride and Prejudice, The Adventures of Huckleberry Finn, Black Beauty
only called the best
Animal Farm, Lord of the Flies, Ulysses, Heart of Darkness, 1984, Catch-22
sold well and called the best
Lord of the Rings, The Great Gatsby, To Kill a Mockingbird, Lolita, Catcher in the Rye
My conclusion: good storytelling makes money; good writing impresses critics.
I remember reading several of the "only called the best" books as assignments in school. I think schools should include more best-sellers in the curriculum. The Great Gatsby and To Kill a Mockingbird are often assigned, but younger readers seem to prefer fantasy, hence the popularity of the Harry Potter books.
Tuesday, April 15, 2025
Another minor mathematical insight: chords and cyclic quadrilaterals - squaring the circle?
Any two intersecting chords of a circle will each get divided into two segments. If one chord has segments of length A and B, and the other chord has segments of length C and D, then A*B = C*D because of the similar triangles that are formed.
Thus, if the diagonals of a quadrilateral have the same property, it must be a cyclic quadrilateral because the diagonals are chords of a circle touching the corners of the quadrilateral.
However, except for a square, the intersection of the diagonals or chords cannot be the center of the circle because that would make two pairs of identical triangles.
Lockhart noted that if one connects the midpoints of the sides of ANY quadrilateral, the result is a parallelogram.
What Lockhart ought to investigate is whether the diagonals of any quadrilateral are chords of a circle. Of course, the answer is no, but why?
The proof is left as an exercise for the reader.
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