Difference between revisions of "1964 AHSME Problems/Problem 39"
(Created page with "==Problem== The magnitudes of the sides of triangle <math>ABC</math> are <math>a</math>, <math>b</math>, and <math>c</math>, as shown, with <math>c\le b\le a</math>. Through int...") |
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label("$c$",Z,dir(110)); | label("$c$",Z,dir(110)); | ||
</asy> | </asy> | ||
| + | |||
| + | ==See Also== | ||
| + | {{AHSME 40p box|year=1964|num-b=38|num-a=40}} | ||
| + | |||
| + | [[Category:Introductory Algebra Problems]] | ||
| + | |||
| + | {{MAA Notice}} | ||
Revision as of 22:19, 24 July 2019
Problem
The magnitudes of the sides of triangle 
are 
, 
, and 
, as shown, with 
. Through interior point 
and the vertices 
, 
, 
, lines are drawn meeting the opposite sides in 
, 
, 
, respectively. Let 
. Then, for all positions of point , 
is less than:
![[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]](http://latex.artofproblemsolving.com/b/8/1/b8175dd073e666965c2dd3700b5b55a1823d0505.png)
See Also
| 1964 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 38 |
Followed by Problem 40 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
