Eureka!

Eureka! is an achievement/trophy and final quest in Fallout: New Vegas if the Courier sides with the NCR in 2281.

Synopsis
The Second Battle of Hoover Dam is erupting as Caesar's Legion storms the dam and attempts to snatch it from New California Republic control. The Courier fights alongside NCR forces to make it to the Legion's base of operations and either kill Legate Lanius or force the Legate to surrender.

Detailed walkthrough
After completing You'll Know It When It Happens, at the very moment the Courier might choose to side with the NCR for the final battle (thus completing For the Republic, Part 2), the player character will be automatically taken from Col. Moore's office to General Oliver's compound, where the General will begin a briefing. The Courier is given the chance to wait, as this begins the end quest for the game. If the player chooses to continue, after the briefing, Legion members attack the Hoover Dam and you need to fight your way through the power plant rooms and head east toward their base. You will start in power plant 04.
 * (Optional) Mike Lawson will speak to you. If you wish to follow his suggestion and do the optional part which involves flushing the Legion invaders through the turbines, agree to his suggestion. To complete the optional part of the quest, follow the staircase immediately to the left in power plant 04 then across the room to enter power plant 03. The console is on the second floor of power plant 03, on the southern wall. Passing a Science check (65) allows the player to use the terminal to engage the turbines. Completing this will require manual intervention; return to power plant 04 by the closest door which is right below the console. On the second floor on the east wall, from where you reenter there is an exit to the Dam exterior. The manual release valve you need to complete this part of the quest is along the river to your left between the broken retainer wall, just after the second sandbag wall. Return to power plant 03.

Proceed to power plant 02, then power plant 01, dispatching Legion troops on your way. Climb the stairs on your left in power plant 01, then proceed north along the balcony. Follow the passage along the left, then go up the stairs, and exit via the elevator to the Hoover Dam visitor center. At this point, you will begin to notice assistance appearing from companions and factions you have helped out along the way. Exit via the double doors on the east side to Hoover Dam.

Proceed to the north along the curved walkway above the dam. There is a door hidden in the side of the dam that you can pick with a Lockpick skill of 75 (hard). Enter the Hoover Dam checkpoint. From here, you can check the terminal; if you have a Science skill of 100 and a detonator, you may use the terminal to detonate one group of legions (the blast is centered near the first tower). When you exit on the other side, you will encounter a squad of NCR Veteran Rangers. Their commander will thank you for your support and ask you how to coordinate their assistance. The player can respond with several dialogue choices which lead to the following actions:
 * The Rangers snipe Legion soldiers for you;
 * The Rangers follow you into battle;
 * The Rangers will pave the way for the player (requires a Speech check of 85);
 * Rangers will fall back, providing no assistance.

After crossing the dam, head left along the bridge to the east, and continue, following the road. Enter the gate to the Legate's camp. Head east into the camp, following the path up the side of the ridge. There, you will encounter Legate Lanius. Note that with a Speech or Barter skill of 100, you can pass a series of skill checks, and avoid fighting him. The Legate is extremely challenging, and many Legionaries will come out and attack you. If you pass all of the previous speech checks and fail the last one, you get another Speech challenge [85] which can convince the Legate to fight you one on one - all the other Legion troops will become friendly to you.
 * There is another tree of answers in dialogue with Lanius - the speech checks that suggest bluffing. This way is actually easier, because it does not require level 100 Speech (checks are 60-75-90-90). In the last Speech check all three answers give the same result, but after it you must choose the  answer, and the Legate will flee.

After either killing the Legate, or passing a series of speech checks to end the fight peacefully, you should notice the gate to the camp has closed. Head back to it. The gate will explode and you will meet with General Oliver.

Behind the scenes
"Eureka" is the state motto of California. The word comes from the ancient Greek word, heúrēka, literally meaning "I have found". There is a story of the famous Greek mathematician Archimedes, who shouted "Eureka!" upon making a great discovery and ran naked through the streets of Syracuse.

Bugs
Heureka! ¡Eureka! Эврика! Еврика!