<PRE><B>Question 1121425</B>
&lt;br&gt;
I personally find the midpoint formulas, and the corresponding formulas for dividing a line segment into a ratio a:b, unpleasant to use, and easy to misuse.  While the formulas are valid, it is not easy to understand how they give you the answers.&lt;br&gt;
For me, it is much easier to solve problems like this less formally.&lt;br&gt;
And much more satisfying, because it is easy to see how the results are obtained.&lt;br&gt;
For example, to find the coordinates of p, the midpoint of BC, find how far it is in the x and y directions from B to C, and go half that far.&lt;br&gt;
B to C in the x direction is from -4 to +6, a change of 10; so go halfway: -4+5 = 1.  And B to C in the y direction is from -6 to 0, a change of 6; so go halfway: -6+3 = -3.&lt;br&gt;
The coordinates of P are (1,-3).&lt;br&gt;
Now for the points that divide AP internally and externally in the ratio 2:1.&lt;br&gt;
A to P in the x direction is a change of -1; in the y direction is a change of -7.&lt;br&gt;
If Q is the point that divides AP in the ratio 2:1 internally, then AQ = 2*QP.  That means Q is 2/3 of the way from A to P.  So the x coordinate of Q is 2/3 of the way from 2 to 1: 2 + (2/3)(-1) = 2-2/3 = 4/3. And the y coordinate is 2/3 of the way from 4 to -3: 4 + (2/3)(-7) = 4 - 14/3 = -2/3.&lt;br&gt;
So the coordinates of Q are (4/3,-2/3).&lt;br&gt;
And if R is the point that divides AP in the ratio 2:1 externally, then AR = 2*AP.  Since the changes in x and y from A to P are -1 and -7, the changes from A to R meed to be twice that -- -2 and -14.&lt;br&gt;
So the coordinates of R are x = 2+(-2) = 0 and y = 4+(-14) = -10.</PRE>