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Crazy Geometry

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Edited by Richard Walker, Monday, 15 Aug 2022, 13:04


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Beautiful Maths Problem

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Edited by Richard Walker, Wednesday, 17 Aug 2022, 17:33

Problem 1358 at gogeometry [1] asked for a proof that in a regular 12-sided polygon the four diagonal shown all meet at a point. This is quite surprising; it\'s not hard to find threee diagonals the meet at a single point but four is rarer.


After playing arouind for a while I found a proof which was reasonably neat, but I had to use sines and cosines at one point, and I'd hoped for something simpler; and there was nothing very illuminating about my proof in any case. Stan Fulger came up with something much more insightful. Here is his beautiful answer.

It uses two well-known facts about triangles.

The altitudes of a triangle, i.e. the lines drawn from each vertex at 90° to the opposite side, meet at a point.

The angle bisectors of a triangle, i.e. the lines which divide each angle in half, meet at a point.

For example



Now for a "look and see" proof.

In the picture below three diagonals are angle bisectors in the blue triangle, so they meet at a point. Also three diagonals are altitudes of the orange triangle and therefore meet at a point. Two of the diagonals are both a bisector in one triangle and an altitude in the other. Therefore all four diagonals meet at a point.


I drew the figures above using GeoGebra classic.

[1] Geometry problem 1358 https://gogeometry.com/school-college/4/p1358-dodecagon-regular-concurrency-diagonal-infographic-classes.htm


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Word of the day - Marshal

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Edited by Richard Walker, Sunday, 14 Aug 2022, 23:13

You can marshal an argument, or you might be a marshal in a sporting event, or you might be a Field Marshal, and there are many other usages but they generally are to do with organising or leading some activity.

Where does the word descend from? Well rather amazingly it originally meant someone who looked after horses. It's from early Germanic *markhaz "horse" + *skalkaz "servant". The asterisks indicate that these words are not actually attested - we don' thave them written down anywhere, so the exact words are a guess. But the first element is like "mare" and a word "scealc" appears in Old English.

The word came to us from Norman French and the French for a farrier (a smith who shoes horses) is still "maréchal-ferrant".

To conclud, here is a rather nice quote from an early printed book, courtesey of the OED.

1474   W. Caxton tr. Game & Playe of Chesse (1883) iii. ii. 85   All maner of werkemen, as goldsmithes, marchallis, smithes of all forges.

Thanks also to RobWords for his excellent YouTube video on military titles.




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Ear Wax

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Edited by Richard Walker, Sunday, 14 Aug 2022, 23:14

I bought some drops for removing earwax. They contain Urea Hydrogen Peroxide...

So you are literally putting urea in your ear.

Permalink 2 comments (latest comment by Richard Walker, Monday, 15 Aug 2022, 14:16)
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Just askin

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Why is Y pronounced ‘Why’?

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Awesome Crossword Clue (not mine)

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Trivial yet extraordinary subject of theory (10) 

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Climbing Plants

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In my garden I’ve got quite a few climbing plants.

Cultivated: Roses, Runner Beans, Sweet Potato, Nicotiana

Wild: Bindweed, Ivy, White Bryony, Blackberry

I’ve always marvelled at climbers and there’s a famous Flanders and Swann song containing the lines

The fragrant honeysuckle spirals clockwise to the sun, 
And many other creepers do the same. 
But some climb anti-clockwise, the bindweed does, for one, 
Or Convolvulus, to give her proper name.

So today I was thinking about climbers and a bit of research came up with this wonderful summary. See what you think.

https://www.theguardian.com/lifeandstyle/2002/apr/13/shopping.gardens2


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Pastry Joke

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How do you persuade a profiterole to go away? Say choux.

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The Thief Of Time

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Most people are just ordinary crastinators but I’m a pro.

Mind you, it’s taken 10,000 hours of putting things off to get where I am.

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Word Of The Day

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lachsschinken: A kind of cured, salted, smoked ham originally from Bavaria. The name derives from the German lachs  + schinken = ‘salmon ham’, from the bright pink colour of the meat. I found the word from today’s Word watch in The Times and what impressed me was the consonant cluster ‘chssch’ in the middle.

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Eeyore Paradox

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Edited by Richard Walker, Tuesday, 2 Aug 2022, 23:43

If I say the best thing to do about Eeyore is ignore him, then I haven’t ignored him, have I?

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Dad Joke

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Edited by Richard Walker, Tuesday, 2 Aug 2022, 21:04

I went into a marquee and hung about a bit. A bloke came up and asked me what I was doing. I said loitering within tent.

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Young Swans at Play

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Edited by Richard Walker, Saturday, 30 Jul 2022, 23:33

My friend Michèle videoed these cygnets playing. I didn't know they did this but as you see they are very lively and quite chaming. This is a YouTube video and if you click on it you can make it full screen in the usual way.


Permalink 3 comments (latest comment by Richard Walker, Monday, 1 Aug 2022, 13:16)
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Knock-knock

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Knock-knock!

Who’s there?

Ash.

Ash who?

Bless you!

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Playground Joke

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What do you call mouse with a poisonous bite?

Venom mouse.

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Vintage aircraft in their hangar last night

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Edited by Richard Walker, Monday, 25 Jul 2022, 23:48

A photo shared by a friend.

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Sweet Potato

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This beautiful flower is a sweet potato. I did not realise that the plant is a vine, related to bindweed and morning glories. Like morning glories each flower lasts only a day.


Permalink 2 comments (latest comment by Richard Walker, Monday, 25 Jul 2022, 23:39)
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Perennial

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When I talk with fellow gardeners the word 'perennial' comes up over and over again.

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Colour joke

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Why is vermi a lot of money?

Cos it’s half vermillion.

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An Elegant Bird

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This elegant little bird, that visited our garden today, is a young moorhen. It was unafraid of us and came really close to take food we put down. We're hoping it will return tomorrow.



Permalink 2 comments (latest comment by Richard Walker, Sunday, 17 Jul 2022, 18:17)
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Topsy-Turvy

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Following my earlier post about CHECKBOOK, I wrote a Python program to find dictionary words with the property they have symmetry about a horizontal mirror line; if you invert them, they remains the same. I used the UK Advanced Cryptics Dictionary, by Ross Beresford (240732 words) as my word source.

Here is the full list of then 352 results. As you might expect some of the words are a wee bit obscure.

['B', 'BE', 'BECK', 'BECKED', 'BED', 'BEDDED', 'BEDE', 'BEDECK', 'BEDECKED', 'BEDIDE', 'BEE', 'BEEB', 'BEECH', 'BI', 'BIB', 'BIBBED', 'BIBCOCK', 'BICE', 'BICKIE', 'BID', 'BIDE', 'BIDED', 'BIKE', 'BIKED', 'BIKIE', 'BIO', 'BIOCIDE', 'BO', 'BOB', 'BOBBED', 'BOBBIE', 'BOCCE', 'BOCHE', 'BOCK', 'BOCKED', 'BOD', 'BODE', 'BODED', 'BODICE', 'BODIED', 'BOH', 'BOK', 'BOKE', 'BOKED', 'BOKO', 'BOO', 'BOOB', 'BOOBED', 'BOOBOO', 'BOOBOOK', 'BOODIE', 'BOODIED', 'BOOED', 'BOOH', 'BOOHED', 'BOOHOO', 'BOOHOOED', 'BOOK', 'BOOKED', 'BOOKIE', 'BOX', 'BOXED', 'C', 'CEDE', 'CEDED', 'CEDI', 'CEE', 'CH', 'CHE', 'CHECK', 'CHECKBOOK', 'CHECKED', 'CHEEK', 'CHEEKED', 'CHI', 'CHIC', 'CHICH', 'CHICHI', 'CHICK', 'CHICO', 'CHID', 'CHIDE', 'CHIDED', 'CHIK', 'CHOC', 'CHOCHO', 'CHOCK', 'CHOCKED', 'CHOCKO', 'CHOCO', 'CHOICE', 'CHOKE', 'CHOKED', 'CHOKO', 'CHOOK', 'CHOOKIE', 'CID', 'COB', 'COBB', 'COBBED', 'COCCI', 'COCCID', 'COCCO', 'COCCOID', 'COCK', 'COCKED', 'COCO', 'COD', 'CODDED', 'CODE', 'CODEBOOK', 'CODED', 'CODEX', 'COED', 'COHO', 'COHOE', 'COKE', 'COKED', 'COO', 'COOED', 'COOEE', 'COOEED', 'COOK', 'COOKED', 'COOKIE', 'COX', 'COXED', 'D', 'DEB', 'DEBBIE', 'DECCIE', 'DECIDE', 'DECIDED', 'DECK', 'DECKED', 'DECKO', 'DECKOED', 'DECO', 'DECODE', 'DECODED', 'DECOKE', 'DECOKED', 'DEE', 'DEED', 'DEEDED', 'DEEK', 'DEICIDE', 'DEID', 'DEKKO', 'DEKKOED', 'DEO', 'DHOBI', 'DI', 'DIB', 'DIBBED', 'DICE', 'DICED', 'DICH', 'DICK', 'DICKIE', 'DID', 'DIDICOI', 'DIDO', 'DIE', 'DIEB', 'DIED', 'DIKE', 'DIKED', 'DIODE', 'DIOXIDE', 'DIXI', 'DIXIE', 'DO', 'DOB', 'DOBBED', 'DOBBIE', 'DOBCHICK', 'DOC', 'DOCK', 'DOCKED', 'DOD', 'DODDED', 'DODO', 'DOE', 'DOEK', 'DOH', 'DOO', 'DOOB', 'DOOK', 'DOOKED', 'E', 'EBB', 'EBBED', 'ECCE', 'ECCO', 'ECHE', 'ECHO', 'ECHOED', 'ECHOIC', 'ECO', 'ECOCIDE', 'ECOD', 'ED', 'EDDIC', 'EDDIE', 'EDDIED', 'EDDO', 'EDH', 'EE', 'EEK', 'EH', 'EHED', 'EKE', 'EKED', 'EO', 'EX', 'EXCEED', 'EXCEEDED', 'EXCIDE', 'EXCIDED', 'EXE', 'EXODE', 'EXODIC', 'H', 'HE', 'HEBE', 'HECH', 'HECK', 'HEED', 'HEEDED', 'HEID', 'HEIDE', 'HEX', 'HEXED', 'HI', 'HIC', 'HICK', 'HICKOK', 'HID', 'HIDE', 'HIDED', 'HIE', 'HIED', 'HIKE', 'HIKED', 'HO', 'HOB', 'HOBO', 'HOBOED', 'HOC', 'HOCK', 'HOCKED', 'HOD', 'HOE', 'HOED', 'HOH', 'HOI', 'HOICK', 'HOICKED', 'HOIK', 'HOIKED', 'HOKE', 'HOKED', 'HOKI', 'HOO', 'HOOCH', 'HOOD', 'HOODED', 'HOODIE', 'HOODOO', 'HOODOOED', 'HOOK', 'HOOKE', 'HOOKED', 'HOX', 'I', 'IBEX', 'IBO', 'ICE', 'ICEBOX', 'ICED', 'ICH', 'ID', 'IDE', 'IDO', 'IKE', 'IO', 'IODIC', 'IODIDE', 'K', 'KEBBIE', 'KEBBOCK', 'KEBOB', 'KECK', 'KECKED', 'KED', 'KEECH', 'KEEK', 'KEEKED', 'KEX', 'KHOIKHOI', 'KIBE', 'KICK', 'KICKED', 'KID', 'KIDD', 'KIDDED', 'KIDDIE', 'KIDDIED', 'KIDDO', 'KIKE', 'KIKOI', 'KO', 'KOB', 'KOBE', 'KOI', 'KOODOO', 'KOOK', 'KOOKED', 'KOOKIE', 'O', 'OB', 'OBECHE', 'OBI', 'OBIED', 'OBO', 'OBOE', 'OCH', 'OCHE', 'OD', 'ODD', 'ODE', 'ODIC', 'OE', 'OH', 'OHIO', 'OHO', 'OI', 'OIK', 'OK', 'OKE', 'OKED', 'OO', 'OOH', 'OOHED', 'O', 'OX', 'OXHIDE', 'OXIDE', 'X', 'XEBEC', 'XI']




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Unique Word

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How is the word

CHECKBOOK

unique? Answer tomorrow.

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Foxgloves in our garden

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Here are the foxgloves we have planted in dappled shade alongside the new fence.



Permalink 2 comments (latest comment by Richard Walker, Thursday, 14 Jul 2022, 15:46)
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Crossing the road joke

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Why did the plank cross the road? Because it was bored.

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Names For Whisky In The Old Westlm

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In the stereotypical Western a moody stranger comes into the saloon and says “Gimme a shot of Red Eye”. The barman, who wears a white apron, slides a glass along the bar, followed by the bottle.

Here are some other Old Western slang names (allegedly at any rate) for whisky.

Tarantula juice

Strychnine

Mountain howitzer

Coffin varnish

Chain lightening

Tanglefoot


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