#black bisector

[squareallworthy]

Triangle Tuesday 3: The orthocenter, the Euler line, and orthocentric systems

Previously, we have looked at two different ways to mark a point in a triangle. First, we drew cevians (lines from the vertices) to the midpoints of the sides and found that they all cross at a point, which is the centroid. Then we tried drawing perpendiculars to the sides from the midpoints, and those all met at the circumcenter. And you could do this with any point on the side of a triangle -- draw a cevian to it, or a perpendicular from it, and see what happens.

This time, though, we're going to do both. That is, we're going to work with the cevians that also form perpendiculars to the sides. These are the altitudes, which run from a vertex to the nearest point on the opposite side, called the foot of the altitude. The three altitudes all meet at a point H, and that's the orthocenter. (The letter H has been used to mark the orthocenter since at least the late 19th century. I believe it's from the German Höhenschnittpunkt, "altitude intersection point.") Anyway, let's prove that the orthocenter exists.

Theorem: the three altitudes of a triangle coincide.

Here's a very simple proof that the three altitudes coincide. It relies on the existence of the circumcenter, which we already proved before. Given a triangle ABC, draw a line through A parallel to the opposite side BC. Do the same at B and C. These lines cross at D, E, and F and form the antimedial triangle (in blue).

Then the altitudes of ABC are also the perpendicular bisectors of DEF. We proved before that perpendicular bisectors all meet at a point, therefore altitudes do as well.

That was easy. Let's do it again, but in a different way. It's not quite as simple, but it includes a large bonus.

Theorem: the three altitudes of a triangle coincide at a point colinear with the circumcenter and centroid, and GH = 2 GO.

Let's take triangle ABC, and let F be the midpoint of side AB. Then mark two points that we already know, the circumcenter O and the centroid G. We'll also draw the median (green) and the perpendicular bisector (blue) that run through F, leaving the other ones out to avoid cluttering the picture.

We already know from our look at the centroid that G cuts segment FC at a third of its length, so GC = 2GF. Let's extend segment OG in the direction of G by twice its length out to a point we'll label H, so that GH = 2GO.

Now consider the two triangles GOF and GHC. By construction, their two blue sides are in the ratio 1:2, and the same for their two black sides. They also meet in vertical angles at their common vertex G. So by side-angle-side, the triangles are similar, and it follows that HC is parallel to OF, and therefore perpendicular to AB. So H lies on the altitude from H to side AB.

By analogous construction, we can show that H also lies on the other two altitudes. So not only have we proved that the altitudes coincide, but also that O, G, and H all lie on a line, and furthermore that G is located one third of the way from O to H, in any triangle.

This proof is due to Leonard Euler, and the line OGH is called the Euler line. Not only these three points but many others as well fall on this line, which we will get to later on.

Let's look at some more properties of the orthocenter and the feet of the altitudes. I'm just going to look at the case of an acute triangle for now, and show how this extends to the obtuse case later.

Theorem: two vertices of a triangle and the feet of the altitudes from those vertices are concyclic.

Proof is easy: the two right triangles AHcC and AHaC share segment AC as a hypotenuse. Therefore AC is a diameter of the common circumcircle of AHcC and AHaC (following from Thales's theorem).

(Incidentally, look at the angle formed by the blue segment and the altitude CHc. It subtends the same arc as angle CAHa, so (by the inscribed angle theorem again) they must be equal. That's not a part of this theorem, so just tuck that fact away for a moment.)

Theorem: a vertex, the two adjacent feet of the altitudes, and the orthocenter are concyclic.

Same idea, but now the right triangles are AHcH and AHbH, and AH is the diameter of the common circumcircle.

(And incidentally, look at the angle formed by the new blue line and the altitude CHc. It subtends the same arc as HbAH, which is same angle as CAHa. So those angles must be equal too. Since both angles between a blue line and the altitude CHc are equal to the same thing, they are equal to each other. Again, not a part of this theorem, just something I wanted to note.)

So those are some interesting concyclicities, but now let's look at the pedal triangle of the orthocenter, which is called the orthic triangle.

Oh, hey, it's made up blue lines, just like the ones we were talking about. And we proved that the two longest blue lines make equal angles with the altitude between them. By symmetry, we can prove the same thing about all the angles made by the blue lines. So that means

Theorem: two sides of the orthic triangle make equal angles with the altitude between them.

Another way to say this is that the altitudes are the angle bisectors of the orthic triangle. And I admit that was kind of a roundabout way to introduce the orthic triangle, but I think it makes the proof of this property easier to follow.

Two other properties of the orthic triangle immediately follow from this:

In an acute triangle, the inscribed triangle with the shortest perimeter is the orthic triangle

and

In an acute triangle, the orthic triangle forms a triangular closed path for a beam of light reflected around a triangle

which are two ways of saying the same thing.

But those two properties only hold for acute triangles. What happens to the orthic triangle in an obtuse triangle? Let's push point C downward to make triangle ABC obtuse and see what happens. To make things clear, I'm going to extend the sides of ABC and the altitudes from line segments into lines. Here's the before:

And here's the after:

The orthocenter has moved outside of triangle ABC, and two of the altitudes have their feet on extensions of the sides of ABC rather than on the segments AC and BC. The orthic triangle now extends outside ABC, and certainly isn't the inscribed triangle with the shortest perimeter any more.

But look at it another way. We now have an acute triangle ABH, and the line HHc is an altitude of both the obtuse triangle ABC and the acute triangle ABH. Meanwhile, lines AC and BC have become altitudes of ABH.

So what we have is essentially the same acute triangle with two swaps: point C trades places with H, and Ha trades places with Hb. That means that our two theorems about concyclic points morph into each other as triangle ABC switches between acute and obtuse. Here's an animation to show the process:

And this is why I didn't bother with the obtuse case above -- each theorem of concyclicity is the obtuse case of the other.

So if we can just exchange the orthocenter with one of the vertices, what does this mean for their relationship? If you are given a group of vertices and lines, how can you tell which one is the orthocenter and which one are the vertices? Well, you can't.

Theorem: Given an acute or obtuse triangle ABC and its orthocenter H, A is the orthocenter of triangle BCH, B is the orthocenter of ACH, and C is the orthocenter of ABH.

The proof comes from consulting either of the "before" and "after" figures above. Take any three lines that form a triangle, red or black. The other three lines are then the altitudes of that triangle. The three feet are where a red and black line meet perpendicularly, so they are the same for all four possible triangles, which means all four share the same orthic triangle.

(Of course, if ABC is a right triangle, then we get a degenerate case, as you can see from the gif at the moment when C and H meet.)

Such an arrangement of four points is called an orthocentric system. Of the four points, one is always located inside an acute triangle formed by the other three, and it's conventional to label the interior one H and the others ABC.

Orthocentric systems pop up all over the place in triangles, so expect to see more of them as we go along. Now, let me do one little lemma about altitudes, and then I'll show something cool about orthocentric systems.

Lemma: the segment of an altitude from the orthocenter to a side of the triangle is equal to the extension of the altitude from the side to the circumcircle.

We can show this with just a little shuffling of angle identities. Extend altitude CHc to meet the circumcircle at C'. The angles CAB and CC'B, labeled in red, subtend the same arcs, so they are equal. Triangle ABHb is a right triangle, so angle HbBA, in blue, is complementary to it. The same is true for the right triangle C'BHc, so the two angles labeled in blue are equal. Then by angle-side-angle, triangles BHcH and HHcC' are congruent, and segment HHc = HcC'.

By the same argument, we can show that triangle AHHc is congruent to AC'Hc, which leads us into the next bit.

Theorem: all the circumcircles of the triangles of an orthocentric system are the same size.

The blue triangle has the same circumcircle as triangle ABC. From the foregoing, the blue and green triangles are congruent. Therefore their circumcircles are the same size as well. The same argument works for ACH and BCH.

So here is an orthocentric system with its four circumcircles.

The four circumcenters O, Oa, Ob, and Oc form another orthocentric system, congruent to the first one.

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.

An index of all posts in this series is available here.

[certified-anakinfucker]

10.11.14.15.16.18.21

(from these swtor asks!)

10: An NPC you love to hate? — Koth. I am shitting on him day in day out for being an annoying asshat. Honestly. Or Lorman, fuck that guy fr. KALIYO DJANNIS KISS MY SWEET ASS CHALLENGE

11: An NPC you hate to love? — [heavy breathing]

14: Favorite attack? — That zeroing-in charge noise on the Sniper’s “Ambush” shot makes me see God tbh, or the Knight/Warrior’s leap attack.

15: Favorite color crystal? — LICKS THE BLACK-PINK CRYSTAL but also that blue/purple/white one.. Tythian Flame.. so many actually oh my god and the gray-red, I could go on

16: Least favorite color crystal? — eh. I believe there is always a crystal to match something but like. Red-Blue, Menacing Glare, Soverign Gold and Blue, Eternal Gold-Blue …… make my eyes hurt blegh

18: Revanite or Black Bisector? — YOU. Hilariously I forgot what the Black Bisector was despite the fact I literally just finished that questline when we were playing. LOL Revanite <333333

21: Favorite planet? — I’m standing outside your door with “The Occupation of Balmorra” blasting from a wholeass speaker system.

[tiredassmage]

17: A quest(line) you loathe doing?

18: Revanite or Black Bisector?

19: Bonus Series quests, yay or nay?

Thank you for the swtor asks!

17. A quest(line) you loathe doing?

Answered here! (In short: Corellia [derogatory]) But, to pick another, let's go... expacs this time: the Gemini Deception is an annoying as all hell chapter. It's got some interesting mechanics to it, I will give it that, but the Gemini Captain is. such an annoying fight. She kicked my ass hard on baby's first time. [It's me, I'm baby.]

18. Revanite or Black Bisector?

Revanites hands down. I honestly... generally forget Black Bisector even exists. It's such a tucked away quest that I so consistently forgot it on most of my characters and I've complacently just... not gone back to it the few times I have.

Honestly, I can be pretty slack about planetary arcs in general these days (the amount of characters I have, after all, is... oof), but the Revanites aren't one I tend to skip because I do actually enjoy it, by and large.

19. Bonus Series quests, yay or nay?

Since I'm so slack about doing planetary quests to begin with, usually nay. I have the vaguest memories of doing them on my first character I think before they... reorganized things and now they no longer show up as part of like the planetary quest progression?

I hoovered up a lot of things on my first character eager to just do all of the things, but I'm more eager to just do the class content at this point, lol. I should revisit some of them though; it was fun sometimes to see follow-ups to what happened in the planetary quests.

[tonkimemphis]

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[swtorpadawan]

When your Gree buddies give you a promotion 😁

You know what? I don’t care that these guys look like the aliens from Independence Day. I love my Gree buddies. 

And hey - we get a promotion from Black Bisector to Red Rhombus! Red Tangents envelope us!

A thousand thanks to @the-sith-in-the-sky-with-diamond​ and gang for carrying me through the dreaded  Terror From Beyond! 

[mc-swtor]

I have officially become the Black Bisector.

[meatbag-status]

“I can’t wait to meet all your friends,” he said.

 “Who the hell was that,” he asked. “A friend,” I said. “Why are all of your friends droids,” he asked. “Because droids are the only loyal things left in this shitty galaxy,“ I replied. “What about SCORPIO,” he inquired. “SCORPIO can do whatever the hell she wants because she’s bae,” I screamed at the top of my lungs.

[darthmarrsgf]

monday night tumblr fic

or: big game hunting for fun and friendship

i love qyzen. i was so jazzed to do his alliance recruitment mission with my consular because we're buds, and this resulted. (more theron/consular. i'm being boring but this is what i have the most written/unpublished for, and also, this is Very Scary to me, so i'm easing into posting, lol.) unbeta'ed as always; probably repeated words without realizing which is one of my biggest personal demons.

~~~

“Caf?”

Sohlara responded with a single raised eyebrow.

“Fine. There’s a little whiskey in there, too.” Theron rolled his eyes as the Jedi huffed out a laugh and turned back to scan the landscape with a pair of downright antique-looking macrobinoculars. “It’s really not as bad as you think.”

She hummed noncommittally. "Should I be worried about you drinking and shooting?”

“Nah. This is just enough to pretend you’re staying warm. Besides, I’ve shot my way out of a fight with a lot more alcohol in me.”

“I can't decide if that’s reassuring or not. Here,” Sohlara said, passing the binocs to Theron. “Due north.”

Theron tucked away the flask as he squinted into the viewfinder. “Big. Think Scorekeeper will approve?”

“We’re about to find out,” she replied, offering Theron a hand after smoothly moving to stand. “We’ll go on foot. If there are more around, I don’t want to draw their attention yet.”

Theron had to pick up his pace to catch up to the woman, who had already outpaced him with seemingly silent strides through the snow. Actually, he reasoned, they probably were silent.

“You probably could have talked him into coming without offering to hunt,” he observed as he reached her side.

She shrugged. “I know I could have. But he’s a respected warrior. It’s only fair that I prove I can regain my score, the same way he did. And hopefully, it will show his men that we're worth helping.”

“So when he calls you Herald—"

“Scorekeeper’s Herald. Qyzen was an old friend of my master’s. I assume you’ve heard that story?”

Theron nodded. Between his close affiliation with several members of the Order, his history in the SIS, and details Sohlara had mentioned herself over the years, Theron was familiar with the unexpected trials that had led to her designation as Barsen’thor.

“He was on Tython when Yuon first fell ill. He helped me without question, but he was captured.”

“Which is bad.” He remembered that much.

“It’s a forfeiture—worse than death. When I freed him, I convinced him that he should try to regain his score by helping me hunt Lord Vivicar.” Sohlara stepped deftly between boulders and snow drifts, moving quickly enough to avoid the beast’s gaze. “He determined that Scorekeeper chose me as her Herald to guide him through a second chance.”

“Well… he wasn’t wrong.”

“I won’t take for granted the trust he gave me, even as a Padawan. A soft thing.” The corner of her mouth quirked. “He has honor. It’s not exactly honor in the same way you or I might see it, but I do my best to respect it.”

“Herald, Barsen’thor, Commander... You’ve racked up quite a reputation.”

“You know what the Esh-ka called me?” She paused, peering around an enormous spire of ice with her binocs.

“I truly have no idea.”

Sohlara leaned toward Theron, expression deadly serious. “Silent Teeth,” she whispered, snapping her jaw shut centimeters from Theron’s face.

“Blast—" Theron flinched, rubbing his ear as Sohlara laughed quietly and turned back to face the tundra.

“They let me pick that one, though. Oh, and there was a group of Gree ambassador droids on Coruscant—I never thought being called a ‘black bisector’ could be such a compliment."

“A black—never mind. I don’t want to know.”

Theron mimicked Sohlara’s slow crouch as they edged their way from behind the boulder. The animal was just yards away, back turned to the pair as it crunched on the bones of some unfortunate, smaller creature with a cracking sound that ricocheted through the icy canyon.

Faster than Theron could blink, she rushed forward on feet bolstered by the momentum of the Force. He sighed—he hated when she did that, because it always took him an extra second to catch up. But he couldn’t help but admire her like this. As much as they both groused about the planet’s climate, Sohlara had seemed particularly at ease since they arrived on Hoth. Between reuniting with her old friend and spending time away from the constant pressures of the Alliance base, the Commander was clearly… lighter.

As he fired off impeccably aimed rounds at the beast, Theron reflected with a twinge of guilt. He should have been paying closer attention to the clear stress Sohlara was exhibiting. She was always getting onto Theron for working too late into the night, but when was the last time she’d been able to snatch more than a few uninterrupted hours to herself?

Theron closely monitored the fight, but Sohlara took the beast down with ease and a particularly theatrical flourish of her saber.

“When was the last time you went on vacation?” Theron asked, slipping his blasters back into their holsters as the Commander wrenched a square of the animal’s pelt from its body with brutal efficiency.

Sohlara blinked up at him, sending a sudden pang of longing through Theron's chest like lightning. He willed himself to commit as much of the moment to memory as he could—the bright pink of her cheeks, brought to the surface by the combination of exertion and Hoth's frigid wind; the strands of chestnut hair flattened against her forehead with sweat; the tingle of awareness at the base of his neck as the protective Force barrier she'd cast around them retreated into her body. Even now, seeing her so vibrant and full of life felt like a miracle after all the time he'd spent trying to forget the way her eyes sparkled when they met his own.

“A vacation? Besides the five years I spent as Arcann’s wall decor, I— No. Sometimes we would stay an extra night to rest if we passed through a big city, but I suppose I've never been on a real vacation.”

Theron stepped forward, letting his fingertips brush her shoulders as he leaned in close to her lips. “Let’s take one. When we get back. Even if it’s just a couple of days.”

Sohlara’s eyes drifted shut, just for a moment, and she swayed into Theron’s space. “What exactly about hunting predator animals for sport on a desolate ice planet inspired this?”

“Nothing to do with the ice ball,” Theron declared, moving out of the way so she could shove the trophy into the sack the hunters had provided. “Although seeing you fight is always sexy. I’m serious, Sohl. Lana is more than capable of handling things for a few days, and we should go while—"

Theron swallowed. He hadn’t meant to say that part out loud, but it was too late now. “While we still have the time.”

Her expression was soft as she turned to face Theron. “Okay,” she murmured, brushing a feather-light kiss over his lips before stepping toward the main trail.

“O— Wait. Okay?” Theron blinked in surprise.

“Okay. Where are you going to take me on vacation, Agent Shan?”

Theron grinned, jogging to close the distance between them. “Somewhere warm. Somewhere private.” He reached forward to brush his hand against her ass, smirking as she nearly stumbled. “Watch it.”

“Watch—" Sohlara jammed her shoulder into his. “You can apologize on our vacation.”

[commanderlurker]

12, 14, 18, 30

Answering these from 10 days ago because I am a winner.

12: Jedi Order or Sith Order, and why?

I started getting REAL DEEP into moral philosophy and aspects of personal and organised religion but stopped because I don’t need that right now. I’ll go with this: lightsabers go whoosh vroom

14: Favorite attack?

Sweeping gunfire is my go to. And I recently discovered XS Freighter Flyby, which I particularly enjoy using when I’m underground because it makes no sense. Above ground though, I imagine a little ‘Yee haw” from Corso whenever I use that attack.

18: Revanite or Black Bisector?

Not gonna lie, I had to do some googling on this. Started watching a video for the Black Bisector mission and I just… drifted off. I think these are a case of Cats saying ‘yeah sure, I’ll get right on that’ and then she never gets right on that.

30: A companion from one class you wish you could've had on another class?

Andronikos in the Smuggler story could be interesting, given the tenuous relationship smugglers and pirates have with each other.

Also, I love Telos from the Inquisitor story. Put him in the Smuggler story and let him dig out relics that the smuggler can sell for Profit.

[rakghoul-legacy]

Tirym Silu Rakghoul Legacy Profile

Name, Race, Class, and Allegiance

Name: Tirym Silu

Race: Rakghoul Twi'lek (Twi'lek In-Game)

Class: Sage

Allegiance:  Rakghoul Alliance

Family, Friends, and Enemies

Family: Huvu Mirr (Wife)

Friends: Sentient Rakghoul Army (Main Force), Galactic Republic (Ally Of Main Forces), Sith Empire (Ally Of Main Forces), Sentient Rakghoul Council (The Rakghoul Council Of Sentience) and Justicars (From In-Game On Coruscant)

Enemies:  Vitiate the Rakghoul Emperor (Master Of Enemies), Xodu Rha (Dark Sided Sith Maurauder Servant to The Rakghoul Emperor), Kenjon Lifshyn (Dark Sided Sniper Servant To Rakghoul Emperor), Brentren (Dark Sided Jedi Knight Servant to The Rakghoul Emperor), Wamau (Dark Sided Scoundrel Servant to The Rakghoul Emperor), Vuyi (Dark Sided Sniper Servant To Starkstar And Love of Starkstar), Cedjess Weigavis (Dark Sided Cyborg SIS Agent [Gunslinger-Based] Servant to Regverso), Xutria (Dark Sided Sith Pureblood Conqueror [Sorcerer-Based] Servant to Xodu Rha), Zragrur (Dark Sided Rattataki Mandolorian [Vanguard-Based] Servant to Wamau), Bubi (Dark Sided Zabrak Makeb Imperial Operative Servant To Xodu Rha), Jusuo (Dark Sided Sith Pureblood THORN Specialist [Sage-Based] Servant To Wamau), Ysixo (Dark Sided Rattataki Kilik Ally [Assassin-Based] Servant to Starkstar), Reejayl Pasktro (Dark Sided Human Constable [Powertech-Based] Servant To Starkstar), Gehli (Dark Sided Sith Pureblood Ensign [Maurauder-Based] Servant To Starkstar), Jaycanna (Dark Sided Cyborg Chosen Of Voss [Sorcerer-Based] Servant To Starkstar), and Ajadar Sabacan (Dark Sided Human High Warlord [Sentinel-Based] Servant To Regverso)

Starting Server, Current Server, Title 1-3 In-Game, Title In-Fiction, and Video Game Story #

Starting Server: Star Forge

Current Server: Star Forge

Title 1 In-Game: Manipulator

Title 2 In-Game: Padawan

Title 3 In-Game: Black Bisector

Legacy Title In-Game: Guardian of Makeb

Title In My Original Fiction: First Force-Sensitive Rakghoul SIS Agent

Video Game Story #: Rakghoul 16

---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Companions

Skitch Zorn = Qyzen Fess Kana = Treek Carsil Boysen = Tharan Cedrax Sapkezi = Zenith LT Houjar Edgbro = LT Felix Iresso Huvu Mirr = Nadia Grell ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Gallery

Tirym Silu Looks 1

Tirym Silu Looks 2

??

Tirym Silu Signature

Tirym Silu X Huvu Mirr According To Dr. Love

Tirym Silu ??

??

[swtorpadawan]

SWTOR question meme: 18 !

SWTOR question meme. Put some numbers in my askbox!

18: Revanite or Black Bisector?

Black Bisector, FTW!

[tishinada]

I don’t remember getting this before from the Gree delegate, Pat-aK:

Me: People always tell me I look good in black.

Pat-aK: From your humanoid face-patterns, I observe you embody deliberate facetiousness, a black bisector survival trait. Exceptional.

So, I interpret this to mean the Gree enclave looks for smart-ass organics to be their “problem solvers”? :D

[mc-swtor]

I never understood these droids. Black sphere? Purple parallel? And apparently I’m a black bisector.

[meatbag-status]

undefined

In-iK, you're freaking my out, but you're also my favorite.

[butmomitwasfortheaesthetic-blog]

tag your aesthetic: math vocab

angle bisector: likes being included, always in the middle, zip up hoodies, oversized shirts, rainy days, big blankets, curly hair

ray: big dreams, never stops, has one best friend, hot chocolate, sunny days, brothers hoodies, sweat pants, day dreams

vertical angles: best friend till the end,emo, beanies, black nail polish, sk8r boy, avoids family, sharpie tattoos, doesnt like you