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With current information available about the Time War, how did the Time Lords end up losing the Time War?
What where the reason or reasons behind the Time Lords losing the Time War so badly, that in the end Galifrey was being besieged by the Daleks to such a degree, that it would seem that only the sky-trenches around the capital were able to hold them off?
The Time Lords didn't 'lose' the Time War - it was effectively halted by The Doctor because he foresaw, not a loss, but a stalemate that would potentially end the entire universe.
The Ninth Doctor talked about the war but didn't reveal the exact details of what happened. When confronted with a Dalek in the episode Dalek, he suggested that they had been responsible for the death of his people:
NINE: That thing killed hundreds of people.
ROSE: It's not the one pointing the gun at me.
NINE: I've got to do this. I've got to end it. The Daleks destroyed my home, my people. I've got nothing left.
However, he didn't actually say they destroyed all his people - just "hundreds". In the episode he also expresses surprise that a single Dalek still exists, showing that he believed the Time War had ended with the complete destruction of all Daleks as well as all Time Lords:
NINE: They're never going to come! Your race is dead! You all burnt, all of you. Ten million ships on fire. The entire Dalek race wiped out in one second.
DALEK: You lie!
NINE: I watched it happen. I made it happen.
It is later confirmed by the Tenth Doctor in The Sound of Drums that he made the decision to end the war, not just destroy the Daleks as Ten had suggested:
TEN: I was the only one who could end it. And I tried. I did. I tried everything.
The Day of The Doctor finally expands on, and in some way alters the outcome of the Time War. In the timeline that The Doctor occupied up until that point, he had ended the Time War which he foresaw could only end in a stalemate between the two sides and the possible destruction of the whole universe. The events of the short prequel Night of the Doctor show that humans had been dragged into the war, and the Sisterhood of Karn were also deeply concerned about the future of all living things.
The Doctor used a weapon called 'The Moment' to wipe out all the Time Lords and Daleks. However, the events of Day of The Doctor result in that past being changed, and the Doctor 'saves' the Time Lords by capturing them inside a bubble universe. As previous stories (beginning with Bad Wolf) revealed that more Daleks had survived the Doctor's original action to end the Time War than he had previously thought, he didn't need to change anything regarding their fate.
