I know the answer to the 'travel around the world' one -- but someone else can give it a shot so I wont give it away

Ok, for anyone interested, the answers to the triangle and dots:

1. Triangle made up of 3 right angles: If you lay out the triangle on a sphere instead of on a flat plane. Heres an exmaple:



Also see here for a better explanation -
http://torus.math.uiuc.edu/jms/java/dragsphere/


2. Using 4 straight lines to cross through 9 dots:



'Think outside the box' was the best way to describe this one.