1964 AHSME Problems/Problem 39

Problem

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:

$\textbf{(A) }2a+b\qquad\textbf{(B) }2a+c\qquad\textbf{(C) }2b+c\qquad\textbf{(D) }a+2b\qquad \textbf{(E) }$ $a+b+c$

$[asy] import math; pair A = (0,0), B = (1,3), C = (5,0), P = (1.5,1); pair X = extension(B,C,A,P), Y = extension(A,C,B,P), Z = extension(A,B,C,P); draw(A--B--C--cycle); draw(A--X); draw(B--Y); draw(C--Z); dot(P); dot(A); dot(B); dot(C); label("$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$",Z,dir(110)); [/asy]$

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1964 AHSME Problems/Problem 39

Problem