What advantages would a triangular shaped spy plane have over a an SR-71?
The 'triangle' wing design that you refer to is pretty common. It is called the delta wing design and is used in a number of high speed aircraft, for example, the North American XB70 Valkyrie.



In fact, the SR 71 wing planform is also a delta, if one removes the engines. One of the reasons is that the wings of the aircrafts are kept inside the Mach cone formed by the aircraft. The Mach angle is given by,

$\mu = sin^{-1} \frac{1}{M}$,

where M is the Mach number.

As the aircraft speed increases, the Mach angle decreases, which means that the aspect ratio should become smaller and smaller. At hypersonic speeds, the delta wing (or a modified one) is the best option as the wing needs to be pretty short for aerodynamics and structural reasons while able to carry the required fuel and generate lift.

The main advantage of the SR-71 Blackbird is its speed, which was about Mach 3.3. It is logical that its replacement should have better speed characteristics. SR 71 used a ramjet during its high speed dash, while its (rumored) successor is expected to use a scramjet engine. 


"Turbo ram scramjet comparative diagram" by GreyTrafalgar - Own work. Licensed under CC BY-SA 3.0 via Commons.

The first one(a) is a turbojet engine, the second one(b) is ramjet, and the third one(c), a scramjet. The ramjet and scramjet engines are similar, though the combustion happens at supersonic speeds in the scramjet engine.

NASA has a hypersonic aircraft for experimental purposes called the X43,which uses a scramjet engine, though the aircraft size is pretty small compared to the A340.


Source: hapb-www.larc.nasa.gov

The closest actual aircraft have come to the shape of the 'Aurora' are the experimental aircraft like the X-37, which has short wings.


"Boeing X-37B inside payload fairing before launch" by US Air Force - http://www.af.mil/News/Photos.aspx?igphoto=2000374856 (direct link). Licensed under Public Domain via Commons.

There are a number of aircraft with triangular wings, most notably almost all European combat aircraft have delta wings.

And by the way, the Lockheed Martin SR 72 Aurora concept looks like this.


"Lockheed Martin SR-72 concept" by Source (WP:NFCC#4). Licensed under Fair use of copyrighted material in the context of Lockheed Martin SR-72" href="//en.wikipedia.org/wiki/File:Lockheed_Martin_SR-72_concept.png">Fair use via Wikipedia.