Russian problems Problem 62 A tangent to the inscribed circle of a triangle drawn parallel to one of the sides meets the other two sides at X and Y. What is the maximum length XY, if the triangle has perimeter p?
Let BC be the side parallel to XY, h the length of the altitude from A, and r the radius of the in-circle. Then XY/BC = (h - 2r)/h. But r.p = h.BC. So XY = (p - 2BC)BC/p = (p 2 /8 - 2(BC - p/4) 2 )/p. So the maximum occurs when BC = p/4 and has value p/8. Russian 61-70 (C) John Scholes jscholes@kalva.demon.co.uk 25 Sep 1998