PUMaC 2014 · 几何（B 组） · 第 6 题

PUMaC 2014 — Geometry (Division B) — Problem 6

[ 6 ] There is a point D on side AC of acute triangle 4 ABC . Let AM be the median drawn from A (so M is on BC ) and CH be the altitude drawn from C (so H is on AB ). Let I be the intersection of AM and CH , and let K be the intersection of AM and line segment BD . We know that AK = 8, BK = 8, and M K = 6. Find the length of AI .

解析

[ 6 ] There is a point D on side AC of acute triangle 4 ABC . Let AM be the median drawn from A (so M is on BC ) and CH be the altitude drawn from C (so H is on AB ). Let I be the intersection of AM and CH , and let K be the intersection of AM and line segment BD . We know that AK = 8, BK = 8, and M K = 6. Find the length of AI . Solution: The line parallel to AM and passing through C meets line BD at some point (call this point 0 E ). Since M is the midpoint of BC , K is the midpoint of EB . Let H be the foot of 0 perpendicular from K onto AB . Since AK = 8 = KB , we see that H bisect AB . Hence 0 AE//KH //CH . Hence AICE is a parallelogram. Thus AI = CE = 2 KM = 12 .