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Triangle $ABC$ has altitudes $\overline{AD},$ $\overline{BE},$ and $\overline{CF}.$ If $AD = 18,$ $BE = 20,$ and $CF$ is a positive integer, then find the largest possible value of $CF.$

 Jun 5, 2024
 #1
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2A = BE * AC

2A = 20 * AC

2A/20 = AC

 

2A = AD * BC

2A = 18 * BC

2A/18 = BC

 

2A  = CF * AB

2A / CF  = AB

 

AC  + BC   >  AB

 

2A/20 + 2A/18  >  2A/CF

 

1/20 + 1/18  > 1/CF

 

38 / 360 > 1/CF

 

19 / 180 > 1/CF

 

CF >  180  / 19  ≈ 9.47

 

Apparently CF  has no max length.....it's min length   = 10

 

Can  someone else verify this ???

 

cool cool cool

 Jun 5, 2024

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