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

Answer 1

Let $h_a, h_b,$ and $h_c$ be the lengths of the altitudes from $A,B,$ and $C$, respectively. We have $\triangle ABE \sim \triangle ACD$, so $\frac{h_b}{h_a} = \frac{h_c}{12}$. Similarly, we have $\frac{h_a}{h_c} = \frac{h_b}{12}$. Multiplying these equations gives $\frac{h_a h_b}{h_c^2} = 1$. By AM-GM, $\frac{h_a h_b}{h_c^2} \le \left(\frac{h_a + h_b}{2h_c}\right)^2$. Substituting $h_a = h_b = 12$ and simplifying yields $\frac{144}{h_c^2} \le \frac{1}{4}$, or $h_c \le 24$. Since $h_c$ must be a positive integer, the largest possible value is $\boxed{24}$.