Proof of Theorem 1

(* ================== A R B I T R A R Y  P =============== *)

Clear[K1,K2,K3,K4,K5,K6,e,f,g,h,j,k];
Clear[s1,s2,s3,s4,s5,s6];
Clear[r1,r2,r3,r4,r5,r6];
Clear[a1,a2,b1,b2,c1,c2];
eq2=K2/K1==a2/a1;
eq3=(K1+K2)/K3==e/f;
eq4=K4/K3==b2/b1;
eq5=(K3+K4)/K5==g/h;
eq6=K6/K5==c2/c1;
r1=K1/s1;
r2=K2/s2;
r3=K3/s3;
r4=K4/s4;
r5=K5/s5;
r6=K6/s6;
S=c2 a2(1/r1-1/r2)+c1 a2(1/r3-1/r4)+c1 a1(1/r5-1/r6)

fixs=
{s1->(1/2)(e+a1+j),
s2->(1/2)(g+a2+j),
s3->(1/2)(g+b1+f),
s4->(1/2)(k+b2+f),
s5->(1/2)(c1+h+k),
s6->(1/2)(c2+h+e)};
Ceva=a1 b1 c1->a2 b2 c2;
fixegk={e->f(c2/c1+a1/a2),g->h(b1/b2+a2/a1),k->j(c1/c2+b2/b1)};
fixK=Solve[{eq2,eq3,eq4,eq5,eq6},{K2,K3,K4,K5,K6}][[1]]

S2=S/.fixs

S3=S2/.fixK//Simplify

num=Numerator[S3]

num2=num/.fixegk//Simplify

num2/.Ceva