1)
BK1||C1A1
Параллельные прямые отсекают на секущих пропорциональные отрезки (теорема Фалеса).
AK/KK1 =AC1/C1B =5/2
CK/KK1= A1C/BA1 =1/2

AK/CK =5 (AC+CK)/CK =5 AC/CK +1 =5 СK/АС =1/4

2)
BA1= 2*A1C
BC= 3*A1C
AC1= 2,5*C1B
AB=3,5*C1B
∠COA1=∠C1OA (вертикальные углы)

sin(CC1B)= sin(180-AC1C) =sin(AC1C)

По теореме синусов:

C1B/sin(BCC1) = BC/sin(CC1B) sin(CC1B)= BC*sin(BCC1)/C1B

AO/sin(AC1C) = AC1/sin(C1OA) AO= AC1*sin(AC1C)/sin(C1OA)
OA1/sin(BCC1) = A1C/sin(COA1) OA1= A1C*sin(BCC1)/sin(C1OA)

AO/OA1= AC1*sin(AC1C)/A1C*sin(BCC1) = 2,5*C1B*3*A1C*sin(BCC1)/A1C*C1B*sin(BCC1) =7,5

sin(BA1A)= sin(180-AA1C) =sin(AA1C)

sin(A1AB)= BA1*sin(BA1A)/AB

CO= A1C*sin(AA1C)/sin(COA1)
OC1= AC1*sin(A1AB)/sin(C1OA)

CO/OC1= A1C*sin(AA1C)/AC1*sin(A1AB) = A1C*3,5*C1B*sin(BA1A)/2,5*C1B*2*A1C*sin(BA1A) =0,7