Question Number 16430 by Tinkutara last updated on 22/Jun/17 $$\mathrm{In}\:\Delta{ABC}\:\mathrm{with}\:\mathrm{usual}\:\mathrm{notation} \\ $$$$\frac{{r}_{\mathrm{1}} }{{bc}}\:+\:\frac{{r}_{\mathrm{2}} }{{ca}}\:+\:\frac{{r}_{\mathrm{3}} }{{ab}}\:\mathrm{is} \\ $$$$\left(\mathrm{1}\right)\:\frac{\mathrm{1}}{{r}}\:−\:\frac{\mathrm{1}}{{R}} \\ $$$$\left(\mathrm{2}\right)\:\frac{\mathrm{1}}{{r}}\:−\:\frac{\mathrm{1}}{\mathrm{2}{R}} \\ $$$$\left(\mathrm{3}\right)\:\frac{\mathrm{1}}{{r}}\:+\:\frac{\mathrm{1}}{\mathrm{2}{R}} \\ $$$$\left(\mathrm{4}\right)\:\frac{\mathrm{1}}{{r}}\:+\:\frac{\mathrm{1}}{{R}} \\ $$…
Question Number 16392 by ajfour last updated on 21/Jun/17 Answered by Tinkutara last updated on 21/Jun/17 Commented by Tinkutara last updated on 21/Jun/17 $$\angle{AMI}\:=\:\mathrm{90}°,\:\angle{MAI}\:=\:\frac{{A}}{\mathrm{2}},\:{s}\:=\:\frac{{a}\:+\:{b}\:+\:{c}}{\mathrm{2}}, \\…
Question Number 16360 by ajfour last updated on 21/Jun/17 Answered by ajfour last updated on 21/Jun/17 $$\:{To}\:{find}\:{x}={CD},\:\:{y}={BD},\:{z}={AD} \\ $$$${BE}\:{is}\:{drawn}\:\bot\:{to}\:{AD},\:{AF}\:{is} \\ $$$${drawn}\:\bot\:{to}\:{AD}\:{produced}. \\ $$$$\:\bigtriangleup{BDE}\:\sim\:\bigtriangleup{ADF}\:\:\left({right}\:\angle\:{and}\right. \\ $$$$\left.\:\:\:\:\:\:\:\:\:\:\:\:{vertically}\:{opposite}\:{angles}\right)…
Question Number 16358 by Tinkutara last updated on 21/Jun/17 $$\mathrm{In}\:\mathrm{any}\:\mathrm{triangle}\:{ABC},\:{a}\:\mathrm{cot}\:{A}\:+\:{b}\:\mathrm{cot}\:{B} \\ $$$$+\:{c}\:\mathrm{cot}\:{C}\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$$$\left(\mathrm{1}\right)\:{r}\:+\:{R} \\ $$$$\left(\mathrm{2}\right)\:{r}\:−\:{R} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{2}\left({r}\:+\:{R}\right) \\ $$$$\left(\mathrm{4}\right)\:\mathrm{2}\left({r}\:−\:{R}\right) \\ $$ Commented by b.e.h.i.8.3.4.1.7@gmail.com…
Question Number 16359 by Tinkutara last updated on 21/Jun/17 $$\mathrm{In}\:\mathrm{a}\:\Delta{ABC}\:\mathrm{if}\:\frac{{s}\:−\:{a}}{{a}\:−\:{b}}\:=\:\frac{{s}\:−\:{c}}{{b}\:−\:{c}}\:,\:\mathrm{then} \\ $$$$\mathrm{prove}\:\mathrm{that}\:{r}_{\mathrm{1}} ,\:{r}_{\mathrm{2}} ,\:{r}_{\mathrm{3}} \:\mathrm{are}\:\mathrm{in}\:\mathrm{A}.\mathrm{P}. \\ $$$$\mathrm{Here}\:{r}_{\mathrm{1}} ,\:{r}_{\mathrm{2}} \:\mathrm{and}\:{r}_{\mathrm{3}} \:\mathrm{are}\:\mathrm{the}\:\mathrm{exradii} \\ $$$$\mathrm{opposite}\:\mathrm{to}\:\mathrm{angles}\:{A},\:{B}\:\mathrm{and}\:{C}\:\mathrm{respectively}. \\ $$ Answered…
Question Number 16354 by Tinkutara last updated on 21/Jun/17 Answered by myintkhaing last updated on 21/Jun/17 $$\frac{{b}^{\mathrm{2}} +{c}^{\mathrm{2}} −{a}^{\mathrm{2}} }{\mathrm{2}{bc}}+\frac{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} −{c}^{\mathrm{2}} }{\mathrm{2}{ab}}=\mathrm{2}−\frac{{c}^{\mathrm{2}} +{a}^{\mathrm{2}} −{b}^{\mathrm{2}}…
Question Number 81884 by rajesh4661kumar@gmail.com last updated on 16/Feb/20 Commented by mind is power last updated on 16/Feb/20 $${applie} \\ $$$${we}\:{can}\:{find}\:{triangle}\: \\ $$$${withe}\:\:\:{sin}^{−} \:{x},{sin}^{−} {y},{sin}^{−}…
Question Number 16338 by ajfour last updated on 20/Jun/17 Commented by mrW1 last updated on 20/Jun/17 $$\mathrm{x}=\frac{\mathrm{ab}\:\mathrm{sin}\:\left(\theta+\emptyset\right)}{\mathrm{a}\:\mathrm{sin}\:\theta\:+\:\mathrm{b}\:\mathrm{sin}\:\emptyset} \\ $$$$\mathrm{y}=\frac{\mathrm{a}\:\sqrt{\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} −\mathrm{2ab}\:\mathrm{cos}\:\left(\theta+\emptyset\right)}\:\mathrm{sin}\:\theta}{\mathrm{a}\:\mathrm{sin}\:\theta+\:\mathrm{b}\:\mathrm{sin}\:\emptyset} \\ $$$$\mathrm{z}=\frac{\mathrm{b}\:\sqrt{\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} −\mathrm{2ab}\:\mathrm{cos}\:\left(\theta+\emptyset\right)}\:\mathrm{sin}\:\emptyset}{\mathrm{a}\:\mathrm{sin}\:\theta+\:\mathrm{b}\:\mathrm{sin}\:\emptyset}…
Question Number 16273 by Tinkutara last updated on 20/Jun/17 $$\mathrm{If}\:\mathrm{in}\:\Delta{ABC}\:{r}_{\mathrm{1}} \:=\:{r}_{\mathrm{2}} \:+\:{r}_{\mathrm{3}} \:+\:{r},\:\mathrm{prove} \\ $$$$\mathrm{that}\:\mathrm{triangle}\:\mathrm{is}\:\mathrm{right}\:\mathrm{angled}. \\ $$ Commented by mrW1 last updated on 20/Jun/17 $$\mathrm{what}\:\mathrm{is}\:\mathrm{r},\mathrm{r}_{\mathrm{1}}…
Question Number 16269 by Tinkutara last updated on 20/Jun/17 $$\mathrm{2}^{\mathrm{nd}} \:\mathrm{part}\:\mathrm{of}\:\mathrm{Q}.\:\mathrm{16214}:\:\mathrm{Prove}\:\mathrm{that} \\ $$$${r}_{\mathrm{1}} \:=\:{s}\:\mathrm{tan}\:\left(\frac{{A}}{\mathrm{2}}\right),\:{r}_{\mathrm{2}} \:=\:{s}\:\mathrm{tan}\:\left(\frac{{B}}{\mathrm{2}}\right), \\ $$$${r}_{\mathrm{3}} \:=\:{s}\:\mathrm{tan}\:\left(\frac{{C}}{\mathrm{2}}\right). \\ $$ Answered by mrW1 last updated…