Difference between revisions of "1988 AIME Problems/Problem 12"
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<cmath>3[(a + 3)(b + 3) + (b + 3)(c + 3) + (c + 3)(a + 3)] = (a+3)(b+3)(c+3)</cmath> | <cmath>3[(a + 3)(b + 3) + (b + 3)(c + 3) + (c + 3)(a + 3)] = (a+3)(b+3)(c+3)</cmath> | ||
<cmath>3(ab + bc + ca) + 18(a + b + c) + 81 = abc + 3(ab + bc + ca) + 9(a + b + c) + 27</cmath> | <cmath>3(ab + bc + ca) + 18(a + b + c) + 81 = abc + 3(ab + bc + ca) + 9(a + b + c) + 27</cmath> | ||
| − | <cmath>9(a + b + c) + 54 = | + | <cmath>9(a + b + c) + 54 = abc=\boxed{441}</cmath> |
| − | |||
== See also == | == See also == | ||
Revision as of 18:17, 10 April 2008
Problem
Let 
be an interior point of triangle 
and extend lines from the vertices through to the opposite sides. Let 
, 
, 
, and 
denote the lengths of the segments indicated in the figure. Find the product 
if 
and 
.
Solution
Call the cevians AD, BE, and CF. Using area ratios (
and 
have the same base), we have:
![$\frac {d}{a + d} = \frac {[PBC]}{[ABC]}$](http://latex.artofproblemsolving.com/6/d/4/6d4ac85ad425625a41e117bcae717d10e8ad5d1a.png)
Similarily, ![$\frac {d}{b + d} = \frac {[PCA]}{[ABC]}$](http://latex.artofproblemsolving.com/4/7/f/47f2d984035fbfe5e48d693e2e43b890e55a66e5.png)
and ![$\frac {d}{c + d} = \frac {[PAB]}{[ABC]}$](http://latex.artofproblemsolving.com/5/4/a/54a5ecd6c3fa66bde1006a9a6a15a21e7b1a234b.png)
.
Then,
![$\frac {d}{a + d} + \frac {d}{b + d} + \frac {d}{c + d} = \frac {[PBC]}{[ABC]} + \frac {[PCA]}{[ABC]} + \frac {[PAB]}{[ABC]} = \frac {[ABC]}{[ABC]} = 1$](http://latex.artofproblemsolving.com/1/0/4/1040a096ae433a59a04030b2b89c285891454e8a.png)
The identity 
is a form of Ceva's Theorem.
Plugging in , we get
![\[\frac{3}{a + 3} + \frac{3}{b + 3} + \frac{3}{c+3} = 1\]](http://latex.artofproblemsolving.com/b/d/3/bd305fd0d0e48ac0b0330975c426104c87c34e3e.png)
![\[3[(a + 3)(b + 3) + (b + 3)(c + 3) + (c + 3)(a + 3)] = (a+3)(b+3)(c+3)\]](http://latex.artofproblemsolving.com/4/0/9/4092762dabdb65624284e3cfe4b786c2f2d9bec3.png)
![\[3(ab + bc + ca) + 18(a + b + c) + 81 = abc + 3(ab + bc + ca) + 9(a + b + c) + 27\]](http://latex.artofproblemsolving.com/5/1/6/5164a27c882483f2b05ae9bff9d3109389210a11.png)
![\[9(a + b + c) + 54 = abc=\boxed{441}\]](http://latex.artofproblemsolving.com/5/8/5/585be2e21d8c019bb5b64a5c1099ff541eef256c.png)
See also
| 1988 AIME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 11 |
Followed by Problem 13 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
