What is algorithm? What formulas, theorems, equations and hints must I use?

Is here some adia? =)

Suppose that answer is XY, where point X lies on CA and Y lies on CB. Then let CX = x, CY = y. Now you have two equations: x + y = P/2, x * y = CA * CB / 2 (P is perimeter). The rest of solution is straightforward.

Thank you very much! Very good adia!

P.S.:
How get x * y = CA * CB / 2  ?
Just simple:
2 * SQUARE_CXY == SQUARE_ABX
2 * 0.5 * x * y * sinXCY == 0.5 * CA * CB * sinXCY
x * y == 0.5 * CA * CB
x * y = CA * CB / 2
I think about it several minute. =)

See http://mathcentral.uregina.ca/mp/previous2011/apr12sol.php for the detailed account of this problem (history, solutions, etc.)