How do we know that Satan, Lucifer, and Devil are the same guy?

Traditionally the fallen angel is referred to as Lucifer ("light-bringing"). And he was named The devil ("slanderer" or "accuser") and Satan ("adversary" or "opposer") after he apostatized. He is called both The devil or Satan in the bible^{Rev 12:7-9}. Whenever the article "The" (Hebrew: ho) is used in-front of devil, it always denotes Lucifer. Just like the example you showed.

The name Lucifer occurs only once in the Hebrew Bible to denote the fallen angel. That is at Isaiah 14:12-15. Comparing its reference to the morning star and fallen angel cast out of heaven with Revelation 12 where the fallen angel is called 'The devil' or 'Satan',^{Rev 12:7-9} we can conclude all three are one and the same.

All other fallen angels are usually given the name demons^{Mat 7:22; Mark 1:34}, and the chief of the demons is called the devil. This must not be taken, however, to indicate that there is a difference of nature with demons and the devil.

Also note that devil is never used in plural form in the bible.

[P.S The name Lucifer was first attributed to the principle fallen angel by St. Jerome. But many church Fathers maintain that Lucifer is not the proper name of the devil, but denotes only the state from which he has fallen. Also note that many early Christians had "Lucifer" as their name and even Jesus was referred as Lucifer in their writings]

And I saw an angel come down from heaven, having the key of the bottomless pit and a great chain in his hand.

And he laid hold on the dragon, that old serpent, which is the Devil, and Satan, and bound him a thousand years,

Revelation 20:1-2

In the Ezekiel passage, though this is God speaking to the King of Tyre, He also speaks to Satan.

Son of man, take up a lament concerning the king of Tyre and say to him: ‘This is what the Sovereign Lord says:

‘You were the seal of perfection, full of wisdom and perfect in beauty. You were in Eden, the garden of God; every precious stone adorned you: carnelian, chrysolite and emerald, topaz, onyx and jasper, lapis lazuli, turquoise and beryl.[b] Your settings and mountings[c] were made of gold; on the day you were created they were prepared. You were anointed as a guardian cherub, for so I ordained you. You were on the holy mount of God; you walked among the fiery stones. You were blameless in your ways from the day you were created till wickedness was found in you.

Ezekiel 28:12-15

The king of Tyre was not in Eden, or a guardian cherub, or on the holy mount of God. This is descriptive of Lucifer, who became Satan.

The Bible says the devil = Satan in Revelation 12:9

So the great dragon was cast out, that serpent of old, called the Devil and Satan, who deceives the whole world; he was cast to the earth, and his angels were cast out with him.

Regarding Lucifer the Bible says in Isaiah 14:12...

How you are fallen from heaven, O Lucifer, son of the morning! How you are cut down to the ground, You who weakened the nations!

So this Lucifer fell from heaven, and Jesus said in Luke 10:18

“I saw Satan fall like lightning from heaven."

Thus Lucifer = Satan. Name changes often denote changes in character in the Bible, thus Lucifer the light bearer (once a good angel) had become Satan (the adversary).

Learn more Bible truth at - http://www.amazingfacts.org/

The problem with the answers previously given here, and what the majority of English-speaking Christians believe, is that it comes from a perspective of the English translation of the Bible. The Bible isn't of English origin. The Old Testament was written in Hebrew (and called Tanach in that language), and the New Testament was written in Greek.

Lucifer isn't the devil. It certainly isn't the name of an angel, fallen or otherwise. The angels of God who are named in the Bible (Michael and Gabriel) have Hebrew names, even in the NT (again, which was written in Greek). The names of angels who war against God are also given by their Hebrew names (satan [Re 12:9] and abaddon [Re 9:11]). In fact, all the names of individuals who are of Hebrew ancestry are given by their Hebrew names in the NT, transliterated (not translated) into Greek, and from Greek into English.

The word Lucifer doesn't even actually occur in the original Bible texts because lucifer is a Latin word (it means light-bearer). The word in the OT translated Lucifer in Is 14:12 is heylel, used only once and derived from halal (1984 of Strong's Hebrew Concordance), a verb meaning to shine.

In Greek, the word for light-bearer is φωσφόρος, transliterated phosphoros, and used once in the NT,

καὶ ἔχομεν βεβαιότερον τὸν προφητικὸν λόγον ᾧ καλῶς ποιεῖτε προσέχοντες ὡς λύχνῳ φαίνοντι ἐν αὐχμηρῷ τόπῳ ἕως οὗ ἡμέρα διαυγάσῃ καὶ φωσφόρος ἀνατείλῃ ἐν ταῖς καρδίαις ὑμῶν

which we have in one English translation as,

We have also a more sure word of prophecy; whereunto ye do well that ye take heed , as unto a light that shineth in a dark place, until the day dawn , and the day star arise in your hearts… – 2 Pe 1:19, KVJ

Lucifer, or more correctly, light-bearer, isn't a name. Like Christ (Anointed One), it's a title. A rank signifying delegated authority, which God bestowed on Adam (along with the armor corresponding to that rank) when He gave him dominion over the creatures of Earth, but not dominion over Earth itself (Ge 1:28). Through Adam's disobedience he surrendered his title (and thus his armor) to the devil. It's when the armor was taken Adam and Eve could see they were naked. (Ps 104:2; Ro 13:12)

Jesus was appointed Anointed One (Christ) after the Baptism of John (giving Jesus authority on Earth in parallel to the lucifer of that time). After His death, He stripped satan of the authority and armor he took from Adam (Luke 11:22), and the keys of death and hell, and rose from the dead. He then entered into the Holiest of All with His Blood, making restitution for Man's sins and cleansing the vessels therein and the armor. The Father then appointed Him lucifer (He 9:23, 24), giving Him dominion over Earth and the creatures of Earth (MT 28:18. Ge 1:26.)

Early Christians certainly didn't believe lucifer was the name of the devil. The Latin translation of the NT uses the Latin word for light-bearer once, for the same verse,

et habemus firmiorem propheticum sermonem cui bene facitis adtendentes quasi lucernae lucenti in caliginoso loco donec dies inlucescat et lucifer oriatur in cordibus vestris, 2 Pe 1:19

The reason why most English-speaking Christians believe the devil is lucifer today? because he knew that by such deceit, few would attempt to challenge him directly, thinking first of all he was given total authority, and that he still had it. But he's a liar, never having had complete authority to begin with, and being completely stripped of any authority he did have over the Earth and man. (Ge 1:28, MT 4:8; 28:18, He 2:14)

In Jesus, the Anointed One (Christ) and Light-Bearer, mankind has been raised higher than Adam was, and the devil is once again roaming the Earth, tempting men individually, but never able to again rule us all.