Question and Answers Forum

All Questions      Topic List

Trigonometry Questions

Previous in All Question      Next in All Question      

Previous in Trigonometry      Next in Trigonometry      

Question Number 27605 by ajfour last updated on 10/Jan/18

Answered by mrW2 last updated on 13/Jan/18

$$\frac{DB}{DC}=\frac{AB}{AC}=\frac{c}{b}$$$$\Rightarrow DB=\frac{ac}{b+c}$$$$\Rightarrow DC=\frac{ab}{b+c}$$$$AD=\frac{2\sqrt{s(s-a)bc}}{b+c}$$$$\frac{AE}{AB}=\frac{AD}{DB}=\frac{2\sqrt{s(s-a)bc}}{b+c}\times \frac{b+c}{ac}=\frac{2\sqrt{s(s-a)bc}}{ac}$$$$\Rightarrow AE=\frac{2c\sqrt{s(s-a)bc}}{2\sqrt{s(a-a)bc}+ac}$$$$$$$$similarly$$$$\Rightarrow AF=\frac{2b\sqrt{s(s-a)bc}}{2\sqrt{s(a-a)bc}+ab}$$$$$$$$a^2=b^2+c^2-2bc\times \cos A$$$$\Rightarrow \cos A=\frac{b^2+c^2-a^2}{2bc}$$$${EF}^2={AE}^2+{AF}^2-2\times AE\times AF\times \cos A$$$$=\frac{4c^{2}s(s-a)bc}{4s(s-a)bc+a^2{c}^2+2ac\sqrt{s(s-a)bc}}+\frac{4b^{2}s(s-a)bc}{4s(s-a)bc+a^2{b}^2+2ab\sqrt{s(s-a)bc}}-2\times \frac{4bcs(s-a)bc}{4s(s-a)bc+a^{2}bc+2a(b+c)\sqrt{s(s-a)bc}}\times \frac{b^2+c^2-a^2}{2bc}$$$$=4s(s-a)bc[\frac{c^2}{4s(s-a)bc+a^2{c}^2+2ac\sqrt{s(s-a)bc}}+\frac{b^2}{4s(s-a)bc+a^2{b}^2+2ab\sqrt{s(s-a)bc}}-\frac{b^2+c^2-a^2}{4s(s-a)bc+a^{2}bc+2a(b+c)\sqrt{s(s-a)bc}}]$$$$=4s(s-a)bc[\frac{c^2}{4s(s-a)bc+a^2{c}^2+2ac\sqrt{s(s-a)bc}}+\frac{b^2}{4s(s-a)bc+a^2{b}^2+2ab\sqrt{s(s-a)bc}}-\frac{b^2+c^2-a^2}{4s(s-a)bc+a^{2}bc+2a(b+c)\sqrt{s(s-a)bc}}]$$$$$$$$\Rightarrow EF=2\sqrt{s(s-a)bc[\frac{c^2}{4s(s-a)bc+a^2{c}^2+2ac\sqrt{s(s-a)bc}}+\frac{b^2}{4s(s-a)bc+a^2{b}^2+2ab\sqrt{s(s-a)bc}}-\frac{b^2+c^2-a^2}{4s(s-a)bc+a^{2}bc+2a(b+c)\sqrt{s(s-a)bc}}]}$$

Commented by ajfour last updated on 13/Jan/18

$$solengthyandyoucoulddoitsir!$$$$thankyouimmensely.$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com