Alice, Bob, and Charlie are in a Love Triangle. Alice loves Bob, but also has feelings for Charlie — or maybe she doesn't, but can't or doesn't want to turn him down (maybe she's even in a relationship with or married to Charlie while pining after Bob). However will she resolve this dilemma? Well, fortunately, she doesn't have to — Charlie meets with a convenient illness, accident, or other such fatal situation, freeing Alice up to go after Bob without guilt. If Charlie is aware of Alice's feelings for Bob, he may tell her with his dying breath that she shouldn't mourn him too much, because he wants his beloved to be happy.
Note that not everybody uses "hypotenuse" in the way it appears here, as a triangle only has one hypotenuse, but two other sides (unless it is equilateral), so it would be logical to call Alice the hypotenuse in the above example. See for instance Tom Lehrer's Lobachevsky, which mentions Metro-Goldwyn-Moskva's production of The Eternal Triangle, "starring Ingrid Bergman as the hypotenuse."
This trope is where The Plot Reaper meets Cleaning Up Romantic Loose Ends. If Charlie's death is not so accidental, it's Murder the Hypotenuse or The Uriah Gambit, depending on whether it's done directly or set up indirectly. See Comforting the Widow. Compare to Ship Sinking.
May still be a Bittersweet Ending and even lead to Dead Guy Junior.
Spoilers may be within.
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Anime and Manga
In No. 6 this happens to Safu. Though it's really more of a case of merging her with a goddess-like entity, thus destroying a huge chunk of her personality, THEN blowing up the building she's in.
In InuYasha, the Inu Yasha - Kikyou - Kagome triangle is finally broken when Naraku engineers Kikyou's death for good, and she ends up perishing peacefully in Inuyasha's arms. Snifffff...
Discussed in Mahou Sensei Negima!. Haruna references how common this happens in Japanese literature when it's revealed that Yue likes Negi in the same way that Nodoka does. The fact that both Yue and Nodoka are standing right behind her as she does this didn't help the situation at all.
It happens midway in From the New World, the hypotenuse who dies Shun is the member of the Love Triangle, that the other two Saki and Satoru are interested in.
In Love Hina, described in volume 9 of the manga, Haruka, Seta and Sarah's mother were a Love Triangle. Seta had promised to choose by the time he turned 30, but Sarah's mother died; Haruka refused to accept victory by default, and rejected Seta for years afterwards. She got over it eventually and they got married.
This is what breaks up the Sibling Triangle between twins Tatsuya and Kazuya and their neighbor Minami in the manga Touch. Right before Kazuya was going to ask Minami's dad for his blessing, too... ouch.
In Yu-Gi-Oh!, Anzu has a Two-Person Love Triangle between Yugi and Yami. At the end of the series, Yami goes to his final rest, having been a pharaoh who was dead for 3,000 years.
Highschool of the Dead: This actually makes things worse for the hero since Rei's now focused on Takashi, but way too often she ends up comparing him to Hisashi, or berating him in a way that makes Takashi think she is. She eventually gets over it and admits that she'd dated Hisashi more because it hurt too much to be with him, but now she's all his.
Psychic Academy: The love triangle between Ai, Orina, and Mew ends with Mew's death. In a subversion of this trope, Ai decides to spend the rest of his life being true to Mew's memory.
Macross Frontier: subverts this. Near the end Sheryl, close to death from her illness pleads Alto to save Ranka, so the two can be together after she dies (emphatised more in novel, but still present in series). Then Ranka heals Sheryl, so the threesome persists, and isn't resoved in series
The movies, meanwhile invert this. The "hypotenuse" is the only one left standing, while her rival is in coma (but probably will make it, though)) and the object of attraction is missing, and probably dead.
Neon Genesis Evangelion: The main love triangle between Shinji, Asuka and Rei ceases to exist when Rei pulls a Heroic Sacrifice to save Shinji, shortly after arriving at the realization that she "wants to be one with him". As a sort-of deconstruction, Asuka ends up doubting the sincerity of Shinji's feelings, accusing him of comming to her because she's basically the only one left, since everyone else (including the clone replacing the Rei we used to know) either scares Shinji or has kicked the bucket at this point.
Many Bleach HitsugayaxMatsumoto fans see Gin's death as this.
More than one Detective Conan case has a person killed for being one of the corners of a Love Triangle or getting in between a potential couple. Much more common if there's an Arranged Marriage (like a Dr. Jerk killing his ex-girlfriend so she won't stand in between him and the daughter of his boss whom he's marrying for the prestige or blackmail (like a policeman shooting his ex girlfriend dead when she attempts to go the Honey Trap way when he's actually about to move on and marry another girl) involved.
And later works and materials tried to present it as Emma being there for Scott after Jean died, which is contradicted by the above moment, since he supposedly lost the feelings he needed comforting over.
In The Searchers, Martin Pawley finds himself in an Accidental Marriage to a Comanche woman, nicknamed Look. Naturally, this gets in the way of his relationship with Laurie Jorgenson. When Martin tries to press Look for information about the killer he's tracking, she panics and runs away. Martin later finds her dead— killed by the US Army in an attack on a Comanche camp.
A tragic example in Crouching Tiger, Hidden Dragon, where Shu Lien's fiancee was killed in battle and also happened to be Mu Bai's best friend and sworn brother. For years, the two of them refused to pursue a relationship with each other, to honour the memory of their friend/fiancee.
First Knight has it with King Arthur's death, leaving Guinevere free to hook up with Lancelot.
Tsu'tey in Avatar. Even though Neytiri only ended up having feelings for Jake, she was technically supposed to have been betrothed to him from the beginning, so to free her from any possible obligation to or conflict with him, the triangle was predictably resolved by redeeming his character and subsequently killing him off.
Cyclops' death in X-Men: The Last Stand can be seen as this crossed with Disposable Fiancé.
Forbidden Planet. Lieutenant Jerry Farman and Commander Adams are in a romantic competition over Altaira. Jerry is killed by the Creature from the Id when it attacks the United Planets ship and Adams and Altaira end up together.
Gone with the Wind: Scarlett has had a crush on Ashley for years, of which he once reciprocated to but then turned her away after he married Melanie. Her crush made her ungrateful for her own doting Rhett. Eventually Melanie dies and her last words seem to be giving Scarlett permission to marry Ashley. By then Rhett has had enough and leaves so she can have Ashley, but she no longer wants him to go. No word after if she marries Ashley.
In Oblivion2013 Victoria is killed less than a day after Julia comes back into the picture. Bonus points comes from it being her own fault, as she inadvertently sics the Drone on herself as well as Jack.
Mandorallen, The Baroness of Vo Ebor and The Baron of Vo Ebor in David Eddings' Belgariad. In the fifth book, the Baron is seriously wounded. In one of the first books of the second pentology, the Malloreon, the Baron dies as a result of his injuries, but Mandorallen and the Baroness don't get together until a new hypotenuse shows up - the Baron's heir. Finally sick of the moping, and the incipient civil war looming between Mandorallen's family and the Baron's, Belgarion storms in and orders Mandorallen and the Baroness to marry and get it over with, paying off the Baron's heir just to make sure everything stays settled.
Patricia Kennealy's The Silver Branch, first book of the Keltiad: The protagonist, Aeron marries Roderick because her parents wish her to. Roderick and her parents die in the same well, murder. She then marries Gwydion, which was what she had wanted but her parents thought a pair of magic-wielding royals to be too scary.
In the novel Mary of Marion Isle by H. Rider Haggard, the wicked accidental wife accidentally drowns allowing the hero to marry to marry the heroine who is his soul mate.
This also means that Snape couldn't have cared less that Lily's son Harry was the target of the entire assassination attempt, maybe even seeing the infant as just another part of the hypotenuse. That's just cold. Dumbledore is infuriated at this.
Semi-averted in Cyrano de Bergerac. After Christian dies, Cyrano has the opportunity to reveal he wrote the letters and finally receive Roxane's love, but he withholds the information so that his beloved Roxane's memory of Christian won't be tarnished.
In Bujold'sKomarr, Imperial Auditor Miles Vorkosigan finds himself falling in love with the wife of an administrator peripherally involved with an apparent terraforming accident (and the feeling is, unbeknownst to him, mutual). Tien Vorsoisson (the administrator in question) turned out to be involved in an embezzlement scam, Ekaterine (the wife in question) decided to leave him over it, and then Tien ended up accidentally killed by the conspirators when he tried to sell them out to Vorkosigan. Too bad things got classified enough to create a whole new set of barriers in any courtship.
This is how the love triangle in The Castle of Otranto (the first Gothic Novel) was resolved, although it's a bit of an unusual variation. While the two surviving sides of the triangle do marry, they're both deeply distraught over the death (as the two love rivals were quite close friends), and it's said that part of the reason they get together is that they know that no one else will fully understand their grief.
In Juliet Marillier's Heir to Sevenwaters, the heroine starts out courting Aidan before realizing she has feelings for Cathal. This dilemma is resolved when the villain randomly kills Aidan.... because he can, preventing Clodagh from having to make a hard decision.
Lolita. Charlotte wants Humbert; Humbert wants Lolita. Charlotte dies in a freak accident.
In Samuel Shellabarger's novel Prince of Foxes, the adventurer Andrea Orsini and the lady Camilla Varano discover an increasing attraction to each other which both resist out of respect for her Cool Old Guy husband Lord Varano. Varano dies heroically during the climactic siege of Città Del Monte, pushing Andrea and Camilla into each others' arms with his dying words.
Middlemarch: Mr Casaubon tries hard to avert this in George Eliot's novel, by forbidding Dorothea from marrying Ladislaw in a codicil to his will, on pain of having her inheritance stripped. Ultimately the plan fails and she marries Ladislaw anyway.
The H.I.V.E. Series: Zero Hour resolves the Otto/Laura/Lucy love triangle by having Lucy heroically Taking the Bullet for Otto soon after their relationship begins to develop. Predictably, in the next book, Otto and Laura's relationship starts to develop, but, less predictably, it turns out Laura has been threatened into turning traitor, delaying their relationship further.
Warrior Cats: Brambleclaw and Ashfur both love Squirrelflight. Subverted in that the hypotenuse dies long after the relationship issue ends.
In Sweet Valley University, Elizabeth breaks up with her long term boyfriend Todd and he begins dating another girl, Gin-Yung. Gin-Yung conveniently contracts a brain tumour and dies, using her last words to tell Todd to get back together with Elizabeth.
Robin Hobb's Realm of the Elderlings series, Burrich is badly injured in a fight against a stone dragon. Despite the fact that there were three separate ways to save his life, only one is attempted and deemed too difficult. He eventually dies, his body thrown into the sea, and Fitz is now able to move in on Burrich's widow Molly, his childhood sweetheart.
In the final book of the Sequel Series of The 39 Clues, Evan, Amy's boyfriend, is killed off for the sole purpose of letting Amy and Jake be together—although she was already cheating on Evan with Jake, anyway. (The entire book was out-of-character for Amy.) The fandom was enraged.
In the Xanth book Isle of View, the characters themselves invoke this—Prince Dolph is engaged to both Nada and Electra, and for various complicated reasons cannot just pick one. With the deadline coming up, Electra and Nada (who get on quite well) quietly start taking turns anytime anything dangerous needs to be done, just in case fate should resolve the problem that way.
Happens to Horatio Hornblower three times over the course of the novels. Hornblower is initially married to Maria, and his Love Interest, the Lady Barbara Wellesy, goes on to marry Admiral Percy Leighton after being spurned by Hornblower. Hownblower is naturally placed under Leighton's command, and is taken prisoner after his ship is disabled in combat. He escapes captivity, and meets Marie, a French widow who helps shelter him, and they start an affair. Hornblower returns to England to learn that Leighton died in battle, and Maria died in childbirth, putting both Hornblower and Barbara in position to marry. A few books later, Marie turns up again, and is killed fighting Napoleon's troops after his return to France.
Dualla has some UST with Lee (which is mostly built up in deleted scenes), but is in a relationship with Billy. He asks her to marry him and she refuses because of her feelings for Lee, then Billy is conveniently killed off by the villain of the week.
Notably an Averted Trope for the most part with any love-triangle involving Lee and Starbuck (and there's a few). His brother's death in the backstory kinda counts as shown in one of the last episodes when a flashback shows Lee and Kara were interested in each other from the moment Zack introduced his fiance to his brother.
In Arrow, there is a Love Triangle between Oliver, Laurel (his ex-girlfriend), and Tommy (his best friend who dated Laurel for a while). In the end of Season 1, Tommy is killed coming to Laurel's rescue while her office building collapses.
Choujin Sentai Jetman: Played with: after a series-long love triangle between Ryu, Kaori and Gai, the "Where Are They Now?" Epilogue shows that Gai is fatally knifed by a bandit on his way to Ryu and Kaori's wedding. However, by this time he had come to accept them as a couple, and makes it to the wedding to congratulate them before he dies. Maria/Rie would also count as this, since her death gives Ryu the closure he needs to finally begin a relationship with Kaori.
Degrassi High: Has a baroque (and rather creepy) example. In the first season, Caitlin dumps Joey because he's rather immature, and because the new kid Claude seems so much more her type. But Claude turns out to be an ass and she dumps him too. In the second season, Claude is thinking of suicide (due to unrelated problems that Caitlin doesn't know about). Desperate for any hope, he asks Caitlin out again, and she tells him to get lost. When she gets home, she finds Claude has sent her flowers and his suicide note. As she's screaming with rage and guilt, her science teacher orders her to tutor her old ex-boyfriend Joey. He's had a lot of Character Development over the course of this season, and comforts her during the tutoring. Soon, they're back together again.
Doctor Who: In "Bad Wolf", the Doctor invites a sweet young woman by the name of Lynda aboard the TARDIS. His regular companion, Rose, who's been in love with him for most of the series, isn't at all impressed by this development. Fortunately for Rose's peace of mind, Lynda doesn't make it to the end of the following episode.
Dollhouse: Mellie loves Paul. Mellie leaves. Paul loves Echo. Echo loves Paul. Mellie returns - Paul loves Mellie too. Mellie dies. Paul and Echo can be together! Well, until he dies.
Has Lavinia die suddenly of Spanish Flu at the end of series two to make way for Matthew/Mary.
Earlier in the series, Vera Bates unexpectedly commits suicide, allowing Bates to marry Anna. Not that this actually makes things any easier, since he's now the prime suspect for her "murder".
Farscape: Gillina meets this fate at the end of season one — the penalty for coming between Official Couple Aeryn and Crichton. Interestingly, so does Crichton. For the same reason.
JAG: An unusual example in "Pilot Error", in that it was the object of both parties' affection, Harm's friend and Annie's husband, rather than one of the competitors, who dies.
Home and Away: Belle was killed off, apparently paving the way for Aden and Nicole.
Justified: Boyd Crowder has liked Ava for quite some time but she was married to his brother Bowman. However, Bowman was an Abusive Spouse and Ava ends up shooting him dead. When he hears what happened, Boyd is quite thrilled that Ava is single once again (he did not really care much for his brother or the way he treated Ava).
MADtv: Parodied in one skit, where two lovers are trapped in the middle of open water, along with their beautiful diving instructor. The second the husband goes with the female diving instructor, Kelly Clarkson, Kelly dies.
Neighbours: Killed off Bridget for this reason and this reason alone. Apparently the writers didn't want to ruin the apparently endless opportunities for a good teenage romance plot by a marriage. It was a storyline that many fans hated because the two were always portrayed as happily married and madly in love.
Noah's Arc: Dre at the end of season 2, freeing up Wade to be with Noah again.
Revolution: In the first season finale, Nora tells Rachel that Miles has feelings for her, despite the fact that Nora and Miles had just hooked up and rekindled an old flame together. Not long after, it's Nora who takes a round to her liver and dies inMiles' arms.
Tayong Dalawa (lit. The Two of Us): Filipino prime time drama, where this is rather overdone. The show's title is ambiguous title. It could either mean the two half-brothers, or one of them and the girl. It gets hyped up in a Tonight Someone Dies preview, and is nicely (in a narrative sense) subverted when the girl dies. This underscores the point of the series, which is about the two brothers putting aside their differences becoming family.
Terra Nova: Josh wants to bring his girlfriend there from the future, even going so far as to make a deal with the Sixers to do so. His pretty female friend Skye decides to help, despite the fact that she obviously likes him. The Sixers manage to get his girlfriend on the next pilgrimage, but she dies immediately after coming through the time fracture when the Phoenix Group sends through a suicide bomber.
Mass Effect 3 can do this with love triangles involving love interests from the first two games, usually by killing off the Mass Effect 2 love interest. Thane automatically dies no matter what, and Miranda, Jack, and Tali can also die, but that's up to the player's actions.
Can also be done with the Mass Effect 1 love interest if it's Ashley or Kaidan that was Shepard's love interest in the original. Just have Shepard be forced to shoot them dead.
Their relationship is never called love, though otherwise is handled with the subtlety of a gorilla wielding a sledgehammer, but in Xenoblade, there's Reyn, Sharla, and Gadolt, wherein it initially appears that Reyn is a Replacement Goldfish to Sharla, but then it's revealed that humans harvested by the Mechon are made into Metal-face units. Gadolt was one, and so Sharla's fiance is back from the dead, complicating the nascent romance between Reyn and Sharla the story has been teasing to this point, but almost as quickly as he enters the plot, Gadolt pulls a Heroic Sacrifice and is out of the picture again.
Averted very notably in Girl Genius, where a major plot arc involved the heroine and her love interest curing the other love interest of a fatal disease (okay, so they did kill him a little bit, but it was more of a rolling death, and he has recovered now).
An earlier hypotenuse, an actor named Lars, did die, but neither of Agatha's primary love interest were ever aware of his existence, much less his removal from the equation.