РЕШЕНИЕ:
S∆ABK + S∆AKM = S∆ / 2 ⇒ S∆ABK = S∆ / 2 – S∆AKM
BK:KM=4:1 ⇒
S∆ABK =
4
S∆AKM
__1
S∆ / 2 – S∆AKM =
4
S∆AKM
__1
1S∆ – 2 S∆AKM = 8 S∆AKM
10 S∆AKM = S∆
S∆AKM = S∆ / 10
S∆ABK = S∆ / 2 – S∆AKM = S∆ / 2 – S∆ / 10 = 4 S∆ / 10 = 2 S∆ / 5
∆AKM ∞ ∆NKB
AM =
KM
BN
__BK
x =
1
BN
__4
BN = 4x / 1
∆ACP ∞ ∆NBP
AC =
PC
BN
__BP
2x__ =
PC
4x / 1
__BP
1 =
PC
2
__BP
S∆ABP =
BP
S∆APC
__PC
S∆ABP =
2
S∆APC
__1
S∆ABP + S∆APC = S∆ ⇒ S∆ABP = S∆ – S∆APC
S∆ – S∆APC =
2
S∆APC
______1
1 S∆ – 1 S∆APC = 2 S∆APC
3 S∆APC = S∆
S∆APC = S∆ / 3
S KPCM = S∆APC – S∆AKM = S∆ / 3 – S∆ / 10 =
7S∆ / 30
S∆BPK = S∆ / 2 – SKPCM = S∆ / 2 – 7S∆ / 30 = 8S∆ / 30 = 4S∆ / 15
S∆BKP / S ∆ABK = 4S∆ / 15 : 2 S∆ / 5 = 4/15 ∙ 5/2 = 2/3
Ответ: 2/3