Question 93
Show that
$$
\sqrt{\frac{|AB_1|}{|AB|}} + \sqrt{\frac{|BC_1|}{|BC|}} + \sqrt{\frac{|CA_1|}{|CA|}} \le \frac{3}{\sqrt{2}}
$$
if the incircle of a triangle $ABC$ touches the sides $BC$, $AC$ and $AB$ at the points $A_1$, $B_1$ and $C_1$, respectively.
$$
\sqrt{\frac{|AB_1|}{|AB|}} + \sqrt{\frac{|BC_1|}{|BC|}} + \sqrt{\frac{|CA_1|}{|CA|}} \le \frac{3}{\sqrt{2}}
$$
if the incircle of a triangle $ABC$ touches the sides $BC$, $AC$ and $AB$ at the points $A_1$, $B_1$ and $C_1$, respectively.