Question 1121425


if {{{B}}}({{{-4}}},{{{-6}}}) and {{{C}}}({{{6}}},{{{0}}}),   the coordinates of {{{P}}}, the midpoint of {{{BC}}} will be

{{{P}}}({{{(-4+6)/2}}},{{{(-6+0)/2}}})

{{{P}}}({{{1}}},{{{-3}}})


since

{{{A}}}({{{x[1]=2}}},{{{y[1]=4}}})
{{{P}}}({{{x[2]=1}}},{{{y[2]=-3}}})

the coordinates of the points which divide {{{AP}}} internally and externally in the ratio {{{2:1}}} are:
use a formula:

if {{{P}}}({{{x[p]}}},{{{y[p]}}}) divides line segment from {{{A}}} to {{{P}}}  in the ratio {{{a:b=2:1}}} {{{internally}}}, then we have:

{{{x[p]=x[1] +(a/(a+b))(x[2]-x[1])}}}

{{{y[p]=y[1] +(a/(a+b))(y[2]-y[1])}}}

if {{{internally}}}, then we have:

{{{x[p]=x[1] +(a/(a-b))(x[2]-x[1])}}}

{{{y[p]=y[1] +(a/(a-b))(y[2]-y[1])}}}

substitute given values:

{{{x[p]=2+(2/(2+1))(1-2)}}}
{{{x[p]=2+(2/3)(-1)}}}
{{{x[p]=2-2/3}}}
{{{x[p]=6/3-2/3}}}
{{{x[p]=4/3}}}

{{{y[p]=4 +(2/(2+1))(-3-4)}}}
{{{y[p]=4 +(2/3)(-7)}}}
{{{y[p]=4 -14/3}}}
{{{y[p]=12/3 -14/3}}}
{{{y[p]= -2/3}}}

and, your point is {{{P}}}({{{4/3}}},{{{-2/3}}})

if divides segment {{{externally}}}:
{{{x[p]=2+(2/(2-1))(1-2)}}}
{{{x[p]=2+(2/1)(-1)}}}
{{{x[p]=2-2}}}

{{{x[p]=0}}}

{{{y[p]=4 +(2/(2-1))(-3-4)}}}
{{{y[p]=4 +(2/1)(-7)}}}
{{{y[p]=4 -14}}}
{{{y[p]= -10}}}


your point is {{{P}}}({{{0}}},{{{-10}}})