<Tri-2025-2222>, hint

Let area of triangle ABC be S

Since AB=A'B (ACA')=2S.  Similarly (CA'C')=4S (CB'C')=2S, (CA'B')= S

Hence (A'B'C')=7S , B" on A'C', B'B" be altitude of A'B'C'. A" on B'C', A'A" be altitude

Let B"B*= 4/7 B"B', through B* parallel to B'C' draw line L'

Since (CA'C')=4S, L pass through point C

Let A"A*=2/7 A"A', through A* parallel to B'C' draw line L'. Since (C'B'C)=2S

L' pass through C too.

Finally Point C is the intersection of line L and L'.