РЕШЕНИЕ:
S∆ABK + S∆AKM = S∆ / 2 ⇒ S∆ABK = S∆ / 2 – S∆AKM
BK:KM=2:11 ⇒
S∆ABK =
2
S∆AKM
__11
S∆ / 2 – S∆AKM =
2
S∆AKM
__11
11S∆ – 22 S∆AKM = 4 S∆AKM
26 S∆AKM = 11S∆
S∆AKM = 11S∆ / 26
∆AKM ∞ ∆NKB
AM =
KM
BN
__BK
x =
11
BN
__2
BN = 2x / 11
∆ACP ∞ ∆NBP
AC =
PC
BN
__BP
2x__ =
PC
2x / 11
__BP
11 =
PC
1
__BP
S∆ABP =
BP
S∆APC
__PC
S∆ABP =
1
S∆APC
__11
S∆ABP + S∆APC = S∆ ⇒ S∆ABP = S∆ – S∆APC
S∆ – S∆APC =
1
S∆APC
______11
11 S∆ – 11 S∆APC = 1 S∆APC
12 S∆APC = 11S∆
S∆APC = 11S∆ / 12
S KPCM = S∆APC – S∆AKM = 11S∆ / 12 – 11S∆ / 26 =
77 S∆ / 156
S∆BPK = S∆ / 2 – SKPCM = S∆ / 2 – 77 S∆ / 156 = S∆ / 156
S∆BKP / S ∆AKM = S∆ / 156 : 11S∆ / 26 = 1/156 ∙ 26/11 = 1/66
Ответ: 1/66