Question Number 78820 by mathocean1 last updated on 20/Jan/20
$$\mathrm{please}\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{fomula}\:\mathrm{to}\: \\ $$$$\mathrm{determinate}\:\mathrm{the}\:\mathrm{equations}\:\mathrm{of}\: \\ $$$$\mathrm{bissectors}\:\mathrm{in}\:\mathrm{triangle}? \\ $$
Answered by MJS last updated on 21/Jan/20
$$\mathrm{bisector}\:\mathrm{between}\:\mathrm{2}\:\mathrm{lines} \\ $$$${l}_{\mathrm{1}} :\:{y}={a}_{\mathrm{1}} {x}+{c}_{\mathrm{1}} \\ $$$${l}_{\mathrm{2}} :\:{y}={a}_{\mathrm{2}} {x}+{c}_{\mathrm{2}} \\ $$$$\mathrm{intersection}: \\ $$$${I}=\begin{pmatrix}{\frac{{c}_{\mathrm{2}} −{c}_{\mathrm{1}} }{{a}_{\mathrm{1}} −{a}_{\mathrm{2}} }}\\{\frac{{a}_{\mathrm{1}} {c}_{\mathrm{2}} −{a}_{\mathrm{2}} {c}_{\mathrm{1}} }{{a}_{\mathrm{1}} −{a}_{\mathrm{2}} }}\end{pmatrix} \\ $$$$\mathrm{direction}\:\mathrm{vectors}: \\ $$$${v}_{\mathrm{1}} =\begin{pmatrix}{\mathrm{1}}\\{{a}_{\mathrm{1}} }\end{pmatrix};\:{v}_{\mathrm{2}} =\begin{pmatrix}{\mathrm{1}}\\{{a}_{\mathrm{2}} }\end{pmatrix} \\ $$$$\mathrm{normalized}: \\ $$$${v}_{\mathrm{1}} =\begin{pmatrix}{\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+{a}_{\mathrm{1}} ^{\mathrm{2}} }}}\\{\frac{{a}_{\mathrm{1}} }{\:\sqrt{\mathrm{1}+{a}_{\mathrm{1}} ^{\mathrm{2}} }}}\end{pmatrix}\:;\:{v}_{\mathrm{2}} =\begin{pmatrix}{\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+{a}_{\mathrm{2}} ^{\mathrm{2}} }}}\\{\frac{{a}_{\mathrm{1}} }{\:\sqrt{\mathrm{1}+{a}_{\mathrm{2}} ^{\mathrm{2}} }}}\end{pmatrix} \\ $$$$\mathrm{direction}\:\mathrm{vectors}\:\mathrm{of}\:\mathrm{bisectors} \\ $$$${v}_{{b}\mathrm{1}} ={v}_{\mathrm{1}} +{v}_{\mathrm{2}} =\begin{pmatrix}{{p}}\\{{q}}\end{pmatrix} \\ $$$${v}_{{b}\mathrm{2}} =\begin{pmatrix}{{q}}\\{−{p}}\end{pmatrix} \\ $$$$\mathrm{bisectors} \\ $$$${l}_{{b}\mathrm{1}} :\:{X}={I}+{tv}_{{b}\mathrm{1}} \\ $$$${l}_{{b}\mathrm{2}} :\:{X}={I}+{tv}_{{b}\mathrm{2}} \\ $$
Commented by jagoll last updated on 21/Jan/20
$$\mathrm{good}\:\mathrm{sir} \\ $$
Commented by mathocean1 last updated on 21/Jan/20
$$\mathrm{hello} \\ $$$$\mathrm{is}\:\mathrm{it}\:\mathrm{the}\:\mathrm{same}\:\mathrm{thing}\:\mathrm{of}\: \\ $$$$\mathrm{bissectrice}\:\mathrm{of}\:\mathrm{angles}? \\ $$
Commented by MJS last updated on 21/Jan/20
$$\mathrm{of}\:\mathrm{course}.\:\mathrm{what}\:\mathrm{is}\:\mathrm{an}\:\mathrm{angle}\:\mathrm{without}\:\mathrm{lines}? \\ $$