Question

Let ABC be a triangle with altitudes h1,h2,h3 and inradius r. Find the minimum value of the expression

(h1+r/ h1-r) + (h2+r / h2-r) +(h3+r / h3-r)

Answer 1

We know that,

$$\begin{gathered} h_1=\frac{2A}{a}andr=\frac{2A}{a+b+c},\text{where A= area of the triangle and a= the base cooresponding to the height}\mathit{\text{h}}1 \\ \text{Now,} \\ \frac{h_1+r}{h_1-r} \\ =\frac{{\displaystyle \frac{2A}{a}}+{\displaystyle \frac{2A}{a+b+c}}}{\frac{2A}{a}-\frac{2A}{a+b+c}} \\ =\frac{a+b+c+a}{a+b+c-a} \\ =\frac{2a+b+c}{b+c} \\ \text{This is minimum when}a=b=c. \\ \text{So, minium value of}\frac{h_1+r}{h_1-r}\text{is}\frac{2a+a+a}{a+a}\text{=}\frac{4a}{2a}\text{=2.} \\ \text{Similarly,}\frac{h_2+r}{h_2-r}\text{=2 and}\frac{h_3+r}{h_3-r}\text{=2} \\ \text{So, the minimum value of}\frac{h_1+r}{h_1-r}\text{+}\frac{h_2+r}{h_2-r}\text{+}\frac{h_3+r}{h_3-r}\text{is 2+2+2=6} \end{gathered}$$