9514 1404 393
Answer:
a, b, e (only)
Step-by-step explanation:
The circle centered at A is all of the points that are distance AB from A.
The circle centered at B is all of the points that are distance AB from B.
The points C and D are points that are distance AB from both A and B.
The line joining those points (C and D) consists entirely of points that are equal distances from A and from B. CD is the perpendicular bisector of AB.
Point M is one of the points on line CD, and is also on segment AB. Because it is on CD, we know it is the same distance from A as from B. That means AM = BM, and M is the midpoint of AB.
__
Points B, C, D are not on the same line, so can form a triangle. As with any triangle, the sum of any two segments is greater than the length of the third. Here, this is expressed as CB + BD > CD.
__
With some work using the Pythagorean theorem, you can show that ...
CM = AM×√3
Similarly, MD = BM×√3.
Adding these two relations tells us ...
CM +MD = √3(AM +BM)
CD = AB×√3 . . . . . AB and CD are not the same length
__
AM and BM are on the same line, so cannot be perpendicular to each other.
_____
Additional comment
When you are studying compass and straightedge constructions, you would do well to actually perform these constructions using those tools on paper. This gives you a feel for what is going on and helps you remember the steps. A computer screen is no substitute for hands-on.
