Did we ever learn why Lux and Fiora are Rivals?
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http://universe.leagueoflegends.com/en_US/story/champion/fiora/
TL:DR
Fiora shammed House Crownguard by rejecting her arranged marriage to a member of their house on their wedding day. Crownsguard declares Honor Duel to the death against Fiora's father, the head of House Laurent.
Lord Laurent, knowing he doesn't stand a chance, attempts and is caught trying to use a strength sapping poison against Crownsguard's champion, and issentenced for execution. To save their house, Fiora and her father enact an ancient Trial by Combat law against her father, with her and him fighting against one another. Fiora strikes the killing blow and "wins", becomes head of House Laurent and preventing the house from being exiled.
Lux, being a daughter of Crownsguard, is why Fiora considers Lux a rival. Lux, having not paid attention to much to the matter due to having been away in training after being forced into army conscription by her parents at a young age, doesn't realize this and tries being friendly, and is rejected by Fiora.
Well we're likely getting a demacia lore update soon(as in this month),so maybe that will shed some light on it.
My guess though is that Fiora is jealous on Lux being part of the military despite the fact that she is a woman. Which is something that fiora wanted,but never got.
Two rich women competing to be the freatest. It's a tale as old as time.
I think Ralanr has the correct answer. Fiora looks down on Lux's life choices. Fiora values independence, freedom, and the strength to be your own person. She views Lux as a naive, weak-willed puppet of House Crownguard, always doing things just to please the family. How much of this is true remains to be seen.
Well, from my knowledge, Lux dislikes Firoa because she always wanted to be like her, but heritage would never allow it.
just think for a moment what would happen if they ever ran into eachother. i im going to assume that them both being aristocracy (?) in demacia they often do.