HMMT 二月 2010 · 几何 · 第 8 题

英文解析

8. [ 6 ] Let O be the point (0 , 0). Let A, B, C be three points in the plane such that AO = 15, BO = 15,
   and CO = 7, and such that the area of triangle ABC is maximal. What is the length of the shortestside of ABC ?
   Answer: 20 We claim that O should be the orthocenter of the triangle ABC . If O is not on analtitude of 4 ABC , suppose (without loss of generality) that AO is not perpendicular to BC . We can
   Geometry Subject Testrotate A around O , leaving B and C fixed, to make AO perpendicular to BC , which strictly increasesthe area. Therefore, if [ ABC ] is maximal then 4 ABC is an isosceles triangle with orthocenter O andbase AB .
   ODEC
   A B
   Let F be the foot of the perpendicular from C to AB . Since ∠ F OA and ∠ COE are vertical, ∠ F AO =F
   AF CF OF +7
   2 2
   ∠ OCE . Then 4 F AO is similar to 4 F CB , so we have = = , so AF = OF + 7 · OF .
   OF BF AF
   2 2 2
   Since AF = 225 − OF , 2 · OF + 7 · OF − 225 = 0, so OF = 9. Then AF = 12, so AB = 24 and
   BC = 20. Thus, the length of the shortest side of 4 ABC is 20.