Question Number 171558 by Shrinava last updated on 17/Jun/22
![In △ABC , I-incenter ID⊥BC , IE⊥CA , IF⊥AB D∈(BC) , E∈(CA) , F∈(AB) I_a , I_b , I_c -excenters. Prove that: Σ_(cyc) ((EF)/(sin (A/2))) + Π_(cyc) ((EF)/(sin (A/2))) = ((1 + 4r^2 )/R) ∙ [I_a I_b I_c ]](https://www.tinkutara.com/question/Q171558.png)
$$\mathrm{In}\:\:\bigtriangleup\mathrm{ABC}\:,\:\mathrm{I}-\mathrm{incenter} \\ $$$$\mathrm{ID}\bot\mathrm{BC}\:,\:\mathrm{IE}\bot\mathrm{CA}\:,\:\mathrm{IF}\bot\mathrm{AB} \\ $$$$\mathrm{D}\in\left(\mathrm{BC}\right)\:,\:\mathrm{E}\in\left(\mathrm{CA}\right)\:,\:\mathrm{F}\in\left(\mathrm{AB}\right) \\ $$$$\mathrm{I}_{\boldsymbol{\mathrm{a}}} \:,\:\mathrm{I}_{\boldsymbol{\mathrm{b}}} \:,\:\mathrm{I}_{\boldsymbol{\mathrm{c}}} -\mathrm{excenters}.\:\mathrm{Prove}\:\mathrm{that}: \\ $$$$\underset{\boldsymbol{\mathrm{cyc}}} {\sum}\:\frac{\mathrm{EF}}{\mathrm{sin}\:\frac{\mathrm{A}}{\mathrm{2}}}\:\:+\:\:\underset{\boldsymbol{\mathrm{cyc}}} {\prod}\:\frac{\mathrm{EF}}{\mathrm{sin}\:\frac{\mathrm{A}}{\mathrm{2}}}\:\:=\:\:\frac{\mathrm{1}\:+\:\mathrm{4}\boldsymbol{\mathrm{r}}^{\mathrm{2}} }{\mathrm{R}}\:\centerdot\:\left[\mathrm{I}_{\boldsymbol{\mathrm{a}}} \mathrm{I}_{\boldsymbol{\mathrm{b}}} \mathrm{I}_{\boldsymbol{\mathrm{c}}} \right] \\ $$