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Question Number 188320 by normans last updated on 27/Feb/23

if the on △ABC, inner circle radius and outer circle radius r and R show that; r = 4R sin (A/2) ∙ sin (B/2) ∙ sin (C/2)

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\boldsymbol{{if}}\:\boldsymbol{{the}}\:\boldsymbol{{on}}\:\bigtriangleup\boldsymbol{{ABC}},\: \\ $$$$\:\:\boldsymbol{{inner}}\:\boldsymbol{{circle}}\:\boldsymbol{{radius}}\:\boldsymbol{{and}}\:\boldsymbol{{outer}}\:\boldsymbol{{circle}}\:\boldsymbol{{radius}}\:\boldsymbol{{r}}\:\boldsymbol{{and}}\:\boldsymbol{{R}}\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\boldsymbol{{show}}\:\boldsymbol{{that}}; \\ $$$$\:\:\:\:\:\:\boldsymbol{{r}}\:=\:\mathrm{4}\boldsymbol{{R}}\:\boldsymbol{{sin}}\:\frac{\boldsymbol{{A}}}{\mathrm{2}}\:\centerdot\:\boldsymbol{{sin}}\:\frac{\boldsymbol{{B}}}{\mathrm{2}}\:\centerdot\:\boldsymbol{{sin}}\:\frac{\boldsymbol{{C}}}{\mathrm{2}} \\ $$$$ \\ $$

Answered by mr W last updated on 28/Feb/23

(a/(sin A))=(b/(sin B))=(c/(sin C))=2R Δ=((abc)/(4R))=2R^2 sin A sin B sin C r=((2Δ)/(a+b+c)) =((2Δ)/(2R(sin A+sin B+sin C))) =((2R^2 sin A sin B sin C)/(R(sin A+sin B+sin C))) =((2R sin A sin B sin C)/(sin A+sin B+sin C)) =((2R sin A sin B sin C)/(4 cos (A/2) cos (B/2) cos (C/2))) =((16R sin (A/2) cos (A/2) sin (B/2) cos (B/2) sin (C/2) cos (C/2))/(4 cos (A/2) cos (B/2) cos (C/2))) =4R sin (A/2) sin (B/2) sin (C/2)

$$\frac{{a}}{\mathrm{sin}\:{A}}=\frac{{b}}{\mathrm{sin}\:{B}}=\frac{{c}}{\mathrm{sin}\:{C}}=\mathrm{2}{R} \\ $$$$\Delta=\frac{{abc}}{\mathrm{4}{R}}=\mathrm{2}{R}^{\mathrm{2}} \:\mathrm{sin}\:{A}\:\:\mathrm{sin}\:{B}\:\mathrm{sin}\:{C} \\ $$$${r}=\frac{\mathrm{2}\Delta}{{a}+{b}+{c}} \\ $$$$\:\:=\frac{\mathrm{2}\Delta}{\mathrm{2}{R}\left(\mathrm{sin}\:{A}+\mathrm{sin}\:{B}+\mathrm{sin}\:{C}\right)} \\ $$$$\:\:=\frac{\mathrm{2}{R}^{\mathrm{2}} \:\mathrm{sin}\:{A}\:\:\mathrm{sin}\:{B}\:\mathrm{sin}\:{C}}{{R}\left(\mathrm{sin}\:{A}+\mathrm{sin}\:{B}+\mathrm{sin}\:{C}\right)} \\ $$$$\:\:=\frac{\mathrm{2}{R}\:\mathrm{sin}\:{A}\:\:\mathrm{sin}\:{B}\:\mathrm{sin}\:{C}}{\mathrm{sin}\:{A}+\mathrm{sin}\:{B}+\mathrm{sin}\:{C}} \\ $$$$\:\:=\frac{\mathrm{2}{R}\:\mathrm{sin}\:{A}\:\:\mathrm{sin}\:{B}\:\mathrm{sin}\:{C}}{\mathrm{4}\:\mathrm{cos}\:\frac{{A}}{\mathrm{2}}\:\mathrm{cos}\:\frac{{B}}{\mathrm{2}}\:\mathrm{cos}\:\frac{{C}}{\mathrm{2}}} \\ $$$$\:\:=\frac{\mathrm{16}{R}\:\mathrm{sin}\:\frac{{A}}{\mathrm{2}}\:\mathrm{cos}\:\frac{{A}}{\mathrm{2}}\:\:\mathrm{sin}\:\frac{{B}}{\mathrm{2}}\:\mathrm{cos}\:\frac{{B}}{\mathrm{2}}\:\mathrm{sin}\:\frac{{C}}{\mathrm{2}}\:\mathrm{cos}\:\frac{{C}}{\mathrm{2}}}{\mathrm{4}\:\mathrm{cos}\:\frac{{A}}{\mathrm{2}}\:\mathrm{cos}\:\frac{{B}}{\mathrm{2}}\:\mathrm{cos}\:\frac{{C}}{\mathrm{2}}} \\ $$$$\:\:=\mathrm{4}{R}\:\mathrm{sin}\:\frac{{A}}{\mathrm{2}}\:\mathrm{sin}\:\frac{{B}}{\mathrm{2}}\:\mathrm{sin}\:\frac{{C}}{\mathrm{2}} \\ $$

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