Question Number 218393 by Nicholas666 last updated on 09/Apr/25
$$ \\ $$$$\:{let}\:{ABC}\:{be}\:{a}\:{triangle}\:{with}\:{incenter}\:{I}. \\ $$$$\:{prove}\:{that}\:{Ia}\:.\:{Ib}\:.\:{Ic}\:\:\leqslant\:\frac{{abc}}{\mathrm{8}}\:\: \\ $$$$ \\ $$
Answered by mr W last updated on 13/Apr/25
$${seems}\:{wrong}! \\ $$$${example}:\:{a}={b}={c} \\ $$$${IA}={IB}={IC}=\frac{{a}}{\:\sqrt{\mathrm{3}}} \\ $$$${IA}\centerdot{IB}\centerdot{IC}=\left(\frac{{a}}{\:\sqrt{\mathrm{3}}}\right)^{\mathrm{3}} =\frac{{a}^{\mathrm{3}} }{\mathrm{3}\sqrt{\mathrm{3}}}\cancel{\leqslant}\frac{{abc}}{\mathrm{8}} \\ $$