The 2-dimensional coordinates of p are (xp, yp) and those of Q are (xQ, yQ). I am not sure how that might help, but with the information provided that is the best that can be done.
The point with coordinates (p, q) will be rotated to the point with coordinates [(p - q)/sqrt(2), (p + q)/sqrt(2)].
True
zero Half the distance between them would be 4 units; so 3 units from P would not be close enough to Q to be equidistant.
q + p
If p then q is represented as p -> q Negation of "if p then q" is represented as ~(p -> q)
The possible coordinates of the midpoint depend on the coordinates of A and T and these depend on what these two points are and how they are related.If A = (p,q) and T = (r,s ) then the midpoint of AT has coordinates [(p+r)/2, ((q+s)/2].
Suppose the radius is r and the bearings of the two points, P and Q are p and q respectively. Then the coordinates of P are [r*cos(p), r*sin(p)] and the coordinates of Q are [r*cos(q), r*sin(q)]. The distance between these two points can be found, using Pythagoras: d2 = (xq - xp)2 + (yq - yp)2 where xp is the x-coordinate of P, etc.
The point with coordinates (p, q) will be rotated to the point with coordinates [(p - q)/sqrt(2), (p + q)/sqrt(2)].
If the coordinates of a point, before reflection, were (p, q) then after reflection, they will be (-p, q).
If you have the coordinates of two points, say P = (a,b) and Q = (c,d), then slope = (b-d)/(a-c) that is, the difference in the y coordinate of the two points divided by the difference in the x coordinate of the points taken in the same order.
If M P and Q are collinear and MP plus PQ equals MQ then P is between M and Q.
True
It would have been possible to give a simple answer if we could see what proportion of the way from A to B the question was about. But, thanks to the wonderful (not!) browser used by this site, we cannot and so this is a general answer.If the question was about p/q of the way, then the coordinates of the point are [(14*(q-p) + 4*p)/q, (1*(q-p) + 23*p)/q].
Points for "Q" in Scrabble
Suppose you have the points with coordinates (p, q) and (r, s) then, provided p is different from r, the slope of the line is (q - s)/(p - r) = m, say. Then, if (x, y) is any point on the line, (x - s)/(y - r) = m That, after simplification, is the linear equation of the line. This will be a lot simpler when you have numerical values for p, q, r and s rather than work algebraically throughout. If p is not different from r, then the equation is x = p (or r), a vertical line.
coplanar
The coordinates of the mid points are the means of the pairs of coordinates of the two end points. Thus, if P = (px, py, pz) and Q = (qx, qy, qz) then the midpoint of PQ is [(px+qx)/2, (py+qy)/2, (pz+qz)/2]. The result can be simplified for 2-dimensional space, or extended to more dimensions.