Vikings are 9, Void Prisms are range 12.
I can tell you exactly how this battles plays out in sudden death. I've seen it happen way too often and it always ends in terran getting massacred in the air.
1)Battle begins and all short range air and ground units rush ahead and get slaughtered by the combination of air units and anti-air units firing from the ground. This is important because it gives time for all air units to move into range.
2)Vikings have a range of 9, Colossus also have a range of 9. Vikings are forced to move to your front lines to attack the Colossus.
3) Void Prisms begin attacking the Vikings. Because of how the AI works the Vikings will now try to attack the Void Prisms. This forces Vikings to move forward 3 more spaces. This will put them about halfway between your army and the Protoss army being that the Protoss army will generally always be separated from your army by 6 spaces because of their immortals.
4) Vikings are now going to likely be the closest units to the Protoss army by this point. Any Templar that reaches the front line that haven't wasted all of their energy yet will storm the Vikings. I'll admit that Templar are sort of a gamble to use since rarely is a fight just PvT and throwing in the Zerg or a Protoss on the other team can make Templar a bad idea, but even without Templar Protoss can still handily stomp Terran air.
Vikings are now only 3 spaces to the Protoss front line. This means that any Stalkers in the first 3 lines of the Protoss army can attack the vikings, but more importantly any Fade Initiates in the first 7 lines of the Protoss army (aka pretty much all of them) can also attack them.
5) From here the Vikings will likely kill all the Void Prisms, but the have already served their purpose and the damage has been done. The Vikings now have to kill all of the other teams Colossus while being attacked by the other teams ground forces. Assuming the Protoss didn't fail and built enough Fade Initiates it's only a matter of time.
If you throw in other races this is even worse since the other races ground to air are also attacking your vikings, but remember even if you exclude other races Templar become viable to use again and it only takes 2 Storms to destroy all of your vikings if they end up clumped together.
I have played many thousands of games and step 3 is where you fail. I mainly play 1v1 games so I have a pretty good understanding of all the matchups and how to manipulate the SD battlefield. The point is that HT are NOT reliable at taking out air because HT have many counters for all the races.
However let me explain why step 3 will not happen if I was the Terran army. This is due to Ravens. Ravens are so efficient at stopping Void Prisms that they might never be able to do a single point of damage during an SD 1v1 battle. If Void Prisms can not damage Vikings they will not get their attention and Vikings will then prioritize Colossi instead of the Void Prisms since they are closer.
BTW Vikings have 11 range not 9.
For 3v3 games it gets too complicated but I can tell you my most powerful combination that has never been defeated in a team game and never lost any SD battle. This is all after all 6 gas are taken in an epic game. The first 6 gas you just mainly get units to survive and serve as a backbone later.
The Protoss mostly just gets immortals and colossi and hardly any air at all. Never more than 12 Colossi however. Just get as many immortals as possible. The Terran player gets at least 12 banshee, 12 raven, and almost pure vikings after that. No more than 4 siege tanks or so. The Terran army will be providing the majority of the AA for all 3 players so you will build a TON of vikings. The Zerg player will be getting at least 9 swarm guardians and 12+ broodlords. The rest is corruptors and HK whichever has more priority. If enemy for example is going super heavy on air then more Corruptors. If more anti ground then more hydra or HK. You do not need HK right off the bat though. They are basically there to fill up space in between the immortals and serve as meat for the colossi and swarms.
Your melee line needs to be just barely fast and strong enough so that your air does not fly over his forces. It just needs to hold out long enough for broodlords to arrive to completely stop the templar. Your initial templar counter will be ravens and infestors. Ravens will fly forward and get stormed and infestors will neural parasite outcasts forcing templar to storm their own outcasts. Once those are down broodlords should take over protecting your air permanently from storms.
You do not deviate from this unless they do something really strange like build tons of banshee where the Terran could then build Thor for example.
The DS games I have played it is probably around 3000. Many of them against other dedicated DS players. Sufficiently experienced players should never ever have their Vikings threatened by units like HT. Vikings should not be exposed to other AA other than extreme range AA like fades or brutalisks. The main reason vikings are ever exposed to other AA was because they did not build the correct units such as high hp melee units or their front load damage is too low by not having enough siege assault units to allow your units to advance as well. Basically nothing should be getting under your vikings if you play correctly.