They are all marvellous in their monuments, if not as cities (Agra grows shoddier by the year, Delhi more subsumed in the explosion of car ownership). But the triangle really belongs to the old days of travel when roads were poor, facilities limited and the coaches found it convenient to encompass the three places with reasonable hotels in a four- or five- day visit.
The wonder of Rajasthan - and it is a wonder - is that you can travel it yourself by train or with a car and driver at relatively affordable cost. And you can find magical places to stay all around the state. The maharajas, the maharawals, the maharanas and even just the plain rajas have converted their palaces into small hotels, where you can sleep in mirrored bedrooms, dine in trophied halls and enjoy the life of those waited upon.
The Maharana of Udaipur started it with the island Lake Palace Hotel in Udaipur. At Jodhpur - the most colourful and lively of the cities of Rajasthan - the Maharaja has converted his Umaid Bhawan Palace, built by a follower of Edwin Lutyens in the 1920s (like the Rockefeller Center), as a building works to counter an economic depression in the region.
The furniture is all Art Deco from Maples, the public rooms are large enough to house a cathedral and the gardens are clipped like an Edwardian country seat with unreal views of the old Meherangarh Fort on the rock.
Or, if you prefer, you can rest up in the crazy-paved and eccentrically planned bungalows of the Maharajah's cousin near Jodhpur's antique shops, Ajit Bhawan Palace. The very clued-up Maharaja of Jodhpur has also set up a business providing a tented camp of some luxury that moves about to the annual Pushkar camel fair and the Festival at Jaisalmer - an ordered array of old-style tents each with their own shower facilities and lavatory, and an ever-present band of retainers to bring you tea and hot water and even, if necessary, hot-water bottles for the night.
But the real joy, for those of us who have reached the point of enjoying being pampered on holiday, is the veritable explosion of small heritage hotels being opened every year in the state. I can't claim to have tried them all.
But of those we did, our favourite was Deodarh, in the centre of the state: a provincial town palace restored by the owner's two sons, both of whom gave up their work in the cities to save a multi-courtyarded, multi-staircased and multi-terraced ancestral seat of Sybaritic charm and hospitality. We stayed two nights. We could easily have stayed five or more.
There is a sacred cave nearby for meditation and, within easy car reach, you can visit the spectacular white marble 15th-century Jain temple at Ranakpur and the formidable walls of what is the finest of all Indian castles, Kumbhalgarh.
The only citadel never taken by the Mughals, it dominates the surrounding hills like a crouched panther on a crag, its 2.5 miles of ramparts surrounding several dozen temples and a wildlife sanctuary in which wolves still cross your path.
Of the dozens of other fortresses of the old Mewar kingdom, nearly all fell by force, their menfolk riding out to die on the enemy sword, the women of the harem committing sati on bonfires to keep themselves from the hands of the conquerors.
Fantastical? Yes. But then Rajasthan is like that. The north-western borderland of India, the stopping point for the invasions that came from the north and the silk caravans that came from the east, it is different from the rest of India. Which is where your own car and driver come in. Rajasthan, desert to the north and hills to the south, is a vast state of subtle changes, small towns and old ways.
You need to stop and stare and take photographs, hundreds and hundreds of them. And it helps to go where you want to, at your own pace.
One of the most fascinating of all the royal palaces of Rajasthan, the 18th-century water gardens and pleasure pavilions of Deeg, are hardly visited because the complex is off the coach route.
Yet, in their faded grandeur and the glorious 500 fountains , they are as entrancing as anywhere in India. Theme tourism? True. But it is bringing in some money and saving some monuments. For, it has to be said, as fast as the tourist dollar is spent, the environment is being degraded.
Forests are chopped down for firewood, the desert is despoiled by atomic testing and a policy of "greening" and the walls of the great fort of Jaisalmer and the merchant havelis of the towns crumble under the combined pressure of climatic change, pollution and carelessness. It needs to be seen now, because no one can be quite sure of it in the future.
Driving between Jaisalmer and Osiyan, we passed a great spray of light and dark grey feathers - a felled bird terrible in its outstretched destruction. A few hundred yards further on, we had to slow down to avoid a magnificent red-headed bird of prey which was standing by the roadside, oblivious to passing vehicles, its eyes looking this way and that. The Indian King Vulture mates for life.
ADRIAN HAMILTON paid pounds 1,591 per person for a 15-night custom-made tour, booked through Greaves Travel (0171-487 5687). The price included flights, driver and car, guides and accommodation and covered Delhi, Agra, Jodhpur, Deogarh, Jaisalmer and Udaipur. Rooms at Deogargh Palace (00 91 2904 52 777) cost from pounds 30 per night.
Claire Gervat paid pounds 420 for a flight from Heathrow to Bombay via Frankfurt on Lufthansa, booked through Trailfinders (0171-938 3366). She paid pounds 3 a night at the Lion Safari Lodge.
Getting to Sasan Gir Lion Sanctuary is a bit of a trial as it's in the middle of nowhere. There are flights every day from Bombay to Keshod, 90km from Sasan Gir, then frequent buses from there. You can get information on organised tours that take in Sasan Gir from the tourist office. Good web sites include www.gujarattourism.com and http://gujaratonline.com.
Visas: British passport holders require a visa to visit India. If you call the 24-hour visa information service (0891 880800), you will spend a lot of time and money finding out the following:
For a six-month tourist visa, you can apply in person or by post to: the High Commission of India, India House, Aldwych, London WC2B 4NA; or Consulate-General of India, The Spencers, 19 Augustus Street, Jewellery Quarter, Hockley, Birmingham B18 6DS; or Consulate-General of India, Fleming House, 134 Renfrew Street, Glasgow G3 7ST. If applying by post, first send a stamped addressed envelope for a visa application form to the Postal Visa Section at any of the addresses above. "You are advised not to finalise your travel arrangements until your visa has been issued," says the High Commission.