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Triangle ABC has Altitudes AD, BE and CF. If AD = 12, BE = 14 and CF is a positive integer, then find the largest possible value of CF.

So far, I only found

(14*c)/2 = (12*b)/2 = (CF*a)/2

c+b > a

26 > CF

7c = 6b = CF*c/2 = x

x/7 = c, x/6 = b, 2x/CF = a

x/7+ x/6 > 2x/CF

 Jan 2, 2020
 #1
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Find an inequality that relates the sength of the altitudes in a triangle (you may have to consider area and side lengths.

 Jan 2, 2020
 #2
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Let h be the third altitude.  Then by the triangle inequality, h + 12 > 14, h + 14 > 12, and 12 + 14 > h.  The largest integer h that satisfies these inequalities is h = 25.

 Jan 3, 2020
 #3
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I already tried this value before and it is incorrect

Guest Jan 3, 2020
 #4
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Since the altitudes are inside the triangle, the triangle is acute.  This gives us the inequalities 12^2 + 14^2 > h^2, h^2 + 12^2 > 14^2, and h^2 + 14^2 > 12^2.  Then the largest h can be is 18.

 Jan 3, 2020

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