Wheres Seasons Greasons

I have tried for years to discover something, anything, about this card with no success.

Triangle Tuesday 2: The circumcenter, pedal triangles, degeneracy, and what even is a triangle anyway?

The circumcenter is almost as simple an idea as the centroid, which we looked at before. To define it, you start the same way. Take triangle ABC, find the midpoints of the sides Ma, Mb, and Mc. Then instead of drawing lines to the midpoints from the vertices, draw perpendicular lines through the midpoints. These lines all coincide at a point O, which is the center of a circle that you can draw through the vertices. The circle is called the circumcircle, and that's why the point is called the circumcenter.

I say almost as simple, but in a sense the circumcenter is simpler than the centroid, because you could easily discover it by accident in the process of simply finding the midpoints. Drawing that perpendicular line, the perpendicular bisector, is the standard way of finding the midpoint of a line segment. It's covered all the way back in Book 1, Proposition 10 of Euclid's Elements, and it's simply this:

So if you find the midpoint of all three sides of a triangle with this method, you've already identified the circumcenter. But that doesn't prove that the perpendicular bisectors always coincide, nor that their point of crossing is the center of the circumcircle. For that, let's return to Euclid (Elements, book 4, proposition 5). Euclid's proof is very straightforward, and leads nicely into something interesting, so we'll follow that, but I will state the theorem differently.

Theorem: the perpendicular bisectors of a triangle coincide and their point of intersection is the center of a circle that meets all three vertices.

Let ABC be a triangle with midpoints of the sides Ma opposite A, similarly for Mb and Mc. Draw perpendiculars to sides AC and BC from their midpoints to meet at point O. Connect three segments from O to A, B, and C.

Consider the two blue triangles. Their sies AMb and CMb are equal, since Mb is the midpoint of AC. They also have OMb in common. Their angles at Mb are right angles, and therefore equal. So they have two sides and one angle the same, making them congruent, and therefore OA = OC.

The same argument applied to the green triangles shows that OB and OC are equal. By transitivity, OA = OB and O is equidistant from the three vertices. The radii of a circle are all equal, so a circle centered at O passing through A also passes through B and C.

Finally, draw a line from O perpendicular to AB. This creates two white triangles with sides OA and OB equal, side OZ in common, and equal right angles at Z. The two triangles are then congruent and the two sides AZ and BZ are equal. So Z is the midpoint Mc, showing that the perpendicular bisectors all meet.

And the same argument works when ABC is obtuse. The circumcenter lands outside the triangle, and in this coloring the white triangles are no longer white, but all the relationships between the segments are the same.

(What Euclid didn't prove is that the perpendicular bisectors of AC and BC do in fact meet somewhere, that is, that they aren't parallel. It's not difficult, but I'm not going to prove that either, at least not yet, for reasons.)

Let's develop another idea. We located the circumcenter by drawing the perpendicular bisectors, but now consider doing this construction in reverse. That is, pick a point, and then draw perpendiculars to the three sides. The intersection of the perpendicular and the side is called the foot of that point with respect to that side. If you do that with with the circumcenter, the feet are of course the midpoints, but you can find the feet for any point.

And if we connect those three feet, we get a triangle. In this case, the medial triangle, which we have seen before. For a point in general, the triangle formed by its feet is called the pedal triangle of that point. ("Pedal" meaning "related to feet," and yes, that is why a lever operated with your foot is also called a pedal.)

So let's draw the pedal triangle for an arbitrary point, move it around, and see what happens. The point is going to sometimes be outside the triangle, but that's all right. With extended sides (dashed lines) we will still be able to draw a perpendicular to find a foot, no matter where the point is.

So there's something interesting -- the three feet become colinear and the pedal triangle flattens out into a straight line when the point is on the circumcircle. Does that always happen?

Looks like it does! So let's prove that. Below is a drawing of the flattened-out pedal triangle of a point on the circumcircle, all labeled up. I've also added a couple dashed lines to make following the proof easier. What we would like to show is that ∠JKP + ∠PKL = 180°.

We're going to extract some information from this drawing based on two facts: a) in a cyclic quadrilateral (meaning it has all vertices on the same circle), opposite angles sum to 180° and b) if two right triangles have the same hypotenuse, the triangles have the same circumcircle. I'm not going to prove either of those here because this post is long enough already, but both of these results follow straightforwardly from the inscribed angle theorem.

Theorem: For a point P on the circumcircle of a triangle ABC, the feet J, K, and L with respect to ABC are colinear.

Okay. PCBA is a cyclic quadrilateral, so

1) ∠BAP + ∠PCB = 180°.

And ∠BAP is the same as ∠LAP, so

2) ∠LAP + ∠PCB = 180°.

The two triangles AKP and ALP are right triangles with the same hypotenuse (the dashed segment AP), so all four points are on the same circle and ALKP is a cyclic quadrilateral. Therefore,

3) ∠LAP + ∠PKL = 180°,

4) ∠PKL = ∠PCB.

Quadrilateral PKCJ is also cyclic (again because of right triangles sharing the same hypotenuse), so

5) ∠JKP = ∠JCP

by the inscribed angle theorem. ∠PCB is supplemental to ∠JCP, so

6) ∠JKP = 180° - ∠PCB

and then combining 4) and 6),

7) ∠JKP + ∠PKL = ∠PCB + (180° - ∠PCB) = 180°,

which means that the pedal triangle of a point on the circumference of a circle is flattened to a line segment. Can we consider such a figure to be a triangle?

Now we can return to Euclid's omission in the existence proof of the circumcircle. Proving that the perpendicular bisectors aren't parallel is equivalent to proving that no two sides of a triangle are parallel, or that the three vertices of a triangle aren't colinear. Euclid didn't do that, but it's pretty simple, so he could have. And then he would simply have said that such an arrangement of line segments isn't a triangle. Modern geometers working with projective geometry can answer differently, and might say that this is a degenerate triangle, but we haven't gotten into that yet.

Let's do one more thing. We can extend the flattened line segment into a line, called the Simson line, after Robert Simson, who never wrote anything about it. It was actually discovered by William Wallace, but not named for him, because that's how things work in math.

The set of all Simson lines from all points on the circumcircle form an envelope in the shape of a deltoid, the Steiner deltoid, named for Jakob Steiner, who for all I can tell was its actual discoverer.

The deltoid is tangent to the sides of the triangle at three points where the Simson line coincides with the sides. I'll have more to say about this lovely deltoid later, but for now, please just enjoy this gif. It took me several hours to figure out how to make it, so if people reading this could spend a collective several hours staring at it, that would be great.

If you found this interesting, please try drawing some of this stuff for yourself! You can use a compass and straightedge, or software such as Geogebra, which I used to make all my drawings. You can try it on the web here or download apps to run on your own computer here.

happy holiday everyone!

On this day, July 27th in 1987, a single was released that would change the world forever.

It's Rick Astley's debut single, Never Gonna Give You Up!

I agree that the phrase "being normal about [group]" can be used to mean "behaving like a typical person (which is good) with respect to [group]", which I dislike. In fact, while writing the above reply, I was thinking of another common usage of the phrase as meaning "having the correct opinions about [group]", which bothers me even more.

If "normal" is being used to mean "correct, popular among people I respect, typical, admirable, common sense", that is a bad way to use words, because it conflates concepts which are important to distinguish.

However, in this particular context, "normal" can also be read as "everyday, chill, neutral, default, forgettable", which does not strike me as a pernicious usage. If you read it this way, then "being normal about [group]" points at an important aspect of tolerating and respecting the group in question.

This concept of "capable of neutral, casual interactions" is particularly useful when assessing a potential friend (or someone you might invite to a groupchat, or someone whose party you might attend, etc.). In that circumstance, it's usually less relevant what their political beliefs are, how much they know about [group], or how much they care about the welfare of [group] -- what you want to know is whether they can treat you like any other person in the friend group. It is awkward and uncomfortable when the prospective acquaintance has very strong positive feelings about your demographic group, or when they are very concerned about interacting with you respectfully, even though those things are probably good in an abstract sense.

To inquire about this by asking "are they normal about [group]?" is suboptimal because of the ambiguity with other meanings of "being normal about", but it is a way to express something that needs to be expressed, and as such I am sympathetic to it.

Hate how people talk about “being normal” about something. That only applies to like, being weirdly obsessed with something unusual. You can tell me to please be normal about riding a train, or watching an Anne Hathaway movie. Things that I KNOW I’m weird about.

If you’re using it to describe whether someone is a bigot or not, it’s completely incoherent. Bigotry is normal to bigots. When I hear someone say “I’m normal about X group” I don’t assume that means they share my beliefs. I assume that means they’re uncritical about their own.

Is there something I’m missing here??

first time using lasso tool, i had a lot of fun with this

horsethoughtbarn 5 name

if horses werent called horses what do you think they should be called

I don't consider "normal" a desirable or praiseworthy state, so the usage of the word to (ironically) describe unusual obsession tends to rub me the wrong way.

That said, I do think that the application of the term to a person's feelings about minorities is pointing at something real. Having strong and unusual emotions about people you interact with on the basis of their demographics is generally awkward, counterproductive, and destructive of empathy and solidarity, even if the emotions are positive.

Personally, I notice that when I have negative aliefs or inclinations related to a demographic group, they prevent me from perceiving that group as "normal" and "just people" -- I feel like I should "balance it out" with positive evaluations of the group, and end up thinking about whether I am being bigoted more than actually interacting with them as a person.

If you decide how to act towards someone based primarily on their demographics, that is the same mistake as bigots make, even if you treat members of othered minorities unusually well instead of unusually poorly. "Being normal about" a group can mean treating members of that group like normal people and interacting with them without having an unusually strong emotional reaction to their membership in a given demographic.

Hate how people talk about “being normal” about something. That only applies to like, being weirdly obsessed with something unusual. You can tell me to please be normal about riding a train, or watching an Anne Hathaway movie. Things that I KNOW I’m weird about.

If you’re using it to describe whether someone is a bigot or not, it’s completely incoherent. Bigotry is normal to bigots. When I hear someone say “I’m normal about X group” I don’t assume that means they share my beliefs. I assume that means they’re uncritical about their own.

Is there something I’m missing here??

for /-yr/ i like the song La Monture from Notre-Dame de Paris

for the fun-to-say pile: méli-mélo, micmac, assujetissement, eussent été, farfelu

i'm now looking at my list of least favorite french words to pronounce and going "too many r's" for about 40% of them and "skill issue" for most of the rest. some of these are actually very fun to pronounce i just couldn't wrap my tongue around them a year or so ago, but now i can i guess??? so that's very exciting. makes me hope that someday i'll be able to pronounce the rest of them. this is a bit pie in the sky because i really don't see myself ever getting there with procureur du roi but you never know. and luckily the french abolished the monarchy so it's not like i'll ever have to use that phrase in modern conversation.

anyway here are the words i actually love pronouncing now: décaféiné diététicien filleul pneumonie

i now feel normal/neutral about these words that used to be hard for me: automne, condamner douloureux électricité, énergie inférieur, supérieur, etc. itinéraire lourdeur salmonellose sclérose subodorer succincte

words that are definitely within the realm of my current capability but i haven't practiced them enough: bugle hiérarchisation méditerranéen phtisie

words that are still the bane of my existence but i live in hope: [yʁ] plus at least one other r or [y] sound: chirurgie, fourrure, marbrure, moirure, nourriture, ordures, peinturlurer, procureur du roi, prurit, purpurin, sculpture, serrurerie, structure, sulfureux, tournure all words beginning with ur-, hur-, or sur- other difficult sequence of r's and vowels: construire and other -truire verbs; lueur and sueur; utérus too many r's: marbre, martre, meurtre, opprobre, proroger, réfrigérateur, rétrograde, rorqual difficult sequence of vowels and/or semivowels: coopérant, extraordinaire, hémorroïdal, kyrie eleison, météorologique, micro-ordinateur, micro-organisme, mouillure, quatuor, vanillier not pronounced the way i would expect from the spelling: indemne, penta-, punk just hard for some reason: humour

Turnchetta playlist for @lesmisshippingshowdown

This is for @lesmisshippingshowdown which allows fanworks to give extra points in the polls. We are trying to get the very canonical and important pairing of Turning Woman #3 (a chorus member from the 2012 movie) and Musichetta (Joly's never-onscreen girlfriend from the book) onto the next round.

Even if you don't have a clue what Les Miserables is, can you vote Turnchetta here? As a favor? And if you're not sure, maybe this playlist will convince you of their deep canonicity and long-term importance to the fandom. Or just do it for chaos. Either one as long as you do it.

Spotify playlist:

Tracklist

Three Coins in the Fountain - Connie Francis

Musichetta stupidotta, scanzonata, innocente - Commenti Sonori

My Baby Loves A Bunch of Authors - Moxy Fruvous

What's Love Got To Do With It - Tina Turner

You Turn The Screws - Cake

Turn, Turn, Turn - Dolly Parton

Who Will Shoe Your Pretty Little Feet - Tennesse Ernie Ford

The World Spins Madly On - The Weepies

Sunday Bloody Sunday - U2

飛哥跌落坑渠 (Teddy Boy in the Gutter) - 李寶瑩, 鄧寄塵, 鄭君綿 歡場三怪

Turn Around - They Might Be Giants

Tangled Up In Blue

Heartaches by the Number - Cyndi Lauper

Three Times a Lady - Sissel

Let's Face The Music And Dance - Diana Krall

Turn The World Around - Womansong

Liner notes and Youtube links under the cut. (Fanmix liner notes means "write a synopsis of an entire hypothetical musical" right? That's how I've always done it.)

These are largely old standards, which meant I had a range of cover options, and I went with women's covers most of the time. However some of them I couldn't find an exact match on Youtube and Spotify so a few tracks will be different between the two.

Three Coins in the Fountain - Connie Francis

This was the first song I thought of for a Musichetta and Three mix! You can read this either as the three being Musichetta, Joly and Bossuet, and only one of them gets a happy ever after - or you can read it as Musichetta, Three, and one of their other working woman friends, and only one of them ends up marrying rich.

2. Musichetta stupidotta, scanzonata, innocente - Commenti Sonori

We needed an actual musichetta on this mix. The title translates as "Muschetta stupid, carefree, and innocent" - this is her in her early days, working, spending time with the girlfriends of her youth like Three, dreaming of a superb future.

3. My Baby Loves A Bunch of Authors - Moxy Fruvous

Here she is getting as she gets more involved with the students, gets drawn into the artistic world, goes to fancy parties, becomes someone's mistress.

4. What's Love Got To Do With It - Tina Turner

That world of surface romance and semi-transactional sex starts to harden her, even as she has one (two?) boys who delight in her and she in them.

5. You Turn The Screws - Cake

In the full musical version this would be a duet between Three and Musichetta where they are growing apart as she draws further into the political, literary, and bourgeois world of her students and Three commits to staying as she is and they both become scornful of each other's priorities. They see each other in passing around the Corinthe and don't speak. (This is probably happening around the time of the July revolution.)

6. Turn, Turn, Turn - Dolly Parton

And time passes and everyone gets older, and maybe it can go on like this forever but time passes and it won't, but it's always been that way. (This song is a quote from the book of Ecclesiastes which is very good poetry to read when you're disillusioned with the world and not sure what the point of keeping going is when it's just more of the same.)

7. Who Will Shoe Your Pretty Little Feet - Tennessee Ernie Ford

There start to be ominous undertones in Musichetta's world. It feels like July 1830 only somehow not the same. Her sweet boys fuss over her but at the same time start making noises about what she'll do when they're gone (but with very little understanding of what she *will* do if they're gone. She doesn't disillusion them of course.

8. The World Spins Madly On - The Weepies

This song plays while both Musichetta and Three are hold up in their separate apartments across town from each other, hearing the gunshots go off and staying in bed, Musichetta thinking about how she's abandoned her boys to fight without her and Three thinking about how she's let Musichetta get involved in all that without her

9. Sunday Bloody Sunday - U2

They wake up and go down to the Rue Chanvrerie and get blood all over their pretty little feet and their eyes meet while they sing.

10. 飛哥跌落坑渠 (Teddy Boy in the Gutter) - 李寶瑩, 鄧寄塵, 鄭君綿 歡場三怪

This is a reprise of the first song, courtesy of 1960s Catonese cinema which rewrote the lyrics as being about a girl of the town finding her boy stinking and disgusting in the gutter. I think it's supposed to be a scathing parody and he's just drunk and wearing too much perfume, but to the extent of my ability to translate the Cantonese, I think it also works here, as Three and Musichetta find the remains of her boys and Three is scornful of her squeamishness while hiding her compassion for her grief

11. Turn Around - They Might Be Giants

Trauma. They don't deal well with the survivor's guilt.

12. Tangled Up In Blue - Indigo Girls

This is the key to the whole love story, I knew this song in the Indigo Girls cover first, so it's always been a song about start-cross lesbians; they knew each other once, and they weren't even that different in class, but one of them ended up drifting and taking whatever manual work she could to get by, and one committed to spending time with college boys and reading medieval Italian poetry, and they keep coming together and separating again because they can't stay apart but they can't compromise with each other either. This is Three's song for Musichetta (how the specific incidents in the song line up with the plot is up to the person who ends up writing the book.)

13. Heartaches by the Number - Cyndi Lauper

This is Musichetta's song for Three - from her POV Three keeps leaving and breaking her heart while stays still, even though at the time Three left she thought it didn't matter and she didn't care, looking back from this end she can't stand it, and she's determined the next time she sees Three she makes it clear how much it hurt her.

14. Three Times a Lady - Sissel

And here she finally meets up with Three again but instead of pouring out her hurt she ends up pouring out her love instead!

15. Let's Face The Music And Dance - Diana Krall

Well, says Three, the world is awful and nothing we do matters, so we might as well keep trying to make it better (this is Three admitting that she loves Musichetta too, and her boys and their lost causes weren't all wrong.)

16. Turn The World Around

(Couldn't find the version on spotify on Youtube, so this is a random women's community chorus.)

With Musichetta's and Three's views reconciled, they realize that the key is to forget everyone's old grievances and come together in solidarity to make the world better for everyone with everyone's skills and resources together, and it does matter, and they lead the Turning women (who have also all paired off now) in this song instead.

Curtain call!