I am a US resident alien and I own a house in Australia where I emigrated from. It was my principal residence. If I sell the house, do I compute the capital gain for US tax purposes by taking the purchase price and the selling price in Australian dollars, working out the difference to give the profit in Aussie dollars, and then convert that to US dollars at the exchange rate applicable at the date of sale, or do I compute the purchase price using the exchnage rate at the date of purchase, then the selling price using the rate at the date of sale, and taking that difference for the profit for US tax purposes? The two methods yield hugely different answers because the US dollar has fallen so far in the meantime. So I might well sell my house at a loss in Aussie dollar terms, but still be hit with a big US capital gains tax bill, even if I don't convert the proceeds into US dollars.

This is interesting situation.
Normally the sale is reported on schedule D (Form 1040). You give purchase price and sale price. For purchase price use dollar rate on the date of purchase and for sale price use dollar rate on date of sale.
Yes, even if you have loss, you may have profit on schedule D.

You said that it was your principal residence. If you owned your main home for two years and lived for two years in past five years, then you can exclude gain of up to $250,000. If you are eligible to exclude the entire gain, you do not report it on the tax return. Read: Your U.S. Tax Return: Profit From the Sale of Your Home