• # question_answer In a $\Delta ABC$ bisector of $\angle C$ meets the side AB at D and circumcentre at E. The maximum value of $CD\cdot DE$ is equal to A) $\frac{{{a}^{2}}}{4}$                                    B) $\frac{{{b}^{2}}}{4}$ C) $\frac{{{c}^{2}}}{4}$            D) $\frac{{{(a+b)}^{2}}}{4}$

[c]  $AB=c$ $BC=a$ $AC=b$ $AD+DB=AB=c$ $CD\times DE=AD\times BD$ $\frac{AD+DB}{2}\ge \sqrt{AD\times BD}$                     $[\because \,\,\,\,\,AB\ge 4M]$ $\frac{c}{2}\ge \sqrt{CD\times DE}$                   $\Rightarrow \,\,\,\,\,\,\,\,\,\,\,CD\times DE\le \frac{{{c}^{2}}}{4}$