Problem
1964 AHSME Problem 39
The magnitudes of the sides of \triangle ABC are a, b, and c, as shown, with c\le b\le a. Through interior point P and the vertices A, B, C, lines are drawn meeting the opposite sides in A', B', C', respectively. Let s=AA'+BB'+CC'. Then, for all positions of point P, s is less than:
\textbf{(A) }2a+b\qquad \textbf{(B) }2a+c\qquad \textbf{(C) }2b+c\qquad \textbf{(D) }a+2b\qquad \textbf{(E) } a+b+c
![[asy] import math; defaultpen(fontsize(11pt)); pair A = (0,0), B = (1,3), C = (5,0), P = (1.5,1); pair X = extension(B,C,A,P), Y = extension(A,C,B,P), Z = extension(A,B,C,P); draw(A--B--C--cycle); draw(A--X); draw(B--Y); draw(C--Z); dot(P); dot(A); dot(B); dot(C); label("$A$",A,dir(210)); label("$B$",B,dir(90)); label("$C$",C,dir(-30)); label("$A'$",X,dir(-100)); label("$B'$",Y,dir(65)); label("$C'$",Z,dir(20)); label("$P$",P,dir(70)); label("$a$",X,dir(80)); label("$b$",Y,dir(-90)); label("$c$",Z,dir(110)); //Credit to bobthesmartypants for the diagram[/asy]](/static/aops/74d9368bbf99bfff0c3c136a7118ed67985ee5f9.png)
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