Question 175453
If straight line AB is divided by the point such that AC:CB = m:n then the projections of AB on x and y axes will also be divided in the same ratio.


Let the coordinates of the points be denoted by the name of the point with a suffix x or y. Thus coordinates of C are ({{{C[x]}}},{{{C[y]}}}) and so on.


Considering the projection on x axis,
{{{(A[x] - C[x])/(C[x] - B[x]) = m/n}}}
{{{n(A[x] - C[x]) = m(C[x] - B[x])}}}
{{{C[x] = (n*A[x] + m*B[x])/(m+n)}}} ___________ (1)


In a similar way, {{{C[y] = (n*A[y] + m*B[y])/(m+n)}}} ____________ (2)


If the point lies outside the line segment AB then m/n will be negative as AC and CB will be in opposite directions.


Here, {{{A[x] = 8}}}, {{{A[y] = 12}}}, {{{B[x] = -4}}}, {{{B[y] = 0}}} and m/n =  +- 3 (- sign comes because C may be outside the line segment AB as well). Let m = +-3k and n = k (k = arbitrary constant).


Substituting these in (1) we have
{{{C[x] = (k*8 +- 3k*(-4))/(+-3k + k)}}}
Either {{{C[x] = (8k - 12k))/(3k + k)}}} or {{{C[x] = (8k + 12k))/(-3k + k)}}}
Either {{{C[x] = -1}}} or {{{C[x] = -10}}}


Substituting into (2) we have
Either {{{C[y] = 3}}} or {{{C[y] = -6}}}


So the possible set of coordinates of C are (-1, 3) and (-10, -6).


{{{drawing( 300, 300, -15, 15, -15, 15,
  grid(1),
  locate( 8, 12, A ), locate( -4, 0, B ), locate( -1, 3, C ), blue(locate( -10, -6, C )),
  line( 8, 12, -4, 0 ),
  circle( -1, 3, 0.5 ),
  blue(circle( -10, -6, 0.5 ))
  )}}}