altitudes

To find the largest possible value of CF, we need to understand the properties of altitudes in a triangle and how they relate to each other. 

In a triangle, an altitude is a line segment drawn from a vertex perpendicular to the opposite side. Altitudes have several important properties that we can use to solve this problem.

First, let's label the vertices of triangle ABC as A, B, and C, and the altitudes as AD, BE, and CF. We are given that AD = 12 and BE = 12. Our goal is to find the largest possible value of CF.

One important property of altitudes is that they divide the triangle into smaller triangles with similar shapes. In other words, the ratio of the lengths of corresponding sides in these smaller triangles is constant.

Let's consider triangle ADE and triangle BCF. These two triangles share an altitude (CF) and have a common side (BE). Therefore, they are similar triangles.

Using the property of similar triangles, we can set up a proportion between corresponding sides:

AD/BE = CF/CF'

where CF' represents the length of CF in triangle BCF.

Since AD = 12 and BE = 12, we can substitute these values into the proportion:

12/12 = CF/CF'

Simplifying this equation gives us:

1 = CF/CF'

Now, since CF is a positive integer and CF' is also a positive integer (as it represents a length), we can conclude that CF must be equal to CF'.

Therefore, the largest possible value of CF is equal to the largest possible value of CF'.

We are given that CF is a positive integer and it is less than 84. To find the largest possible value of CF', we need to find the largest positive integer less than 84.

The largest positive integer less than 84 is 83. Therefore, the largest possible value of CF is 83.

In summary, the largest possible value of CF in triangle ABC, given that AD = 12, BE = 12, and CF is a positive integer less than 84, is 83.

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