A. Given points A and B and a constant k > 0, use algebra to prove that there is exactly one point C in segment AB so that |AC/BC| = k. Find the value of t, so that the point C = (1-t)A+tB.
B. Given points A and B and a constant k > 0, use algebra to prove that there is exactly one point D in on line AB but not in segment AB so that |AD/BD| = k. [This is true except for one special case of k. What is this k?] Find the value of t, so that the point D = (1-t)A+tB.
This figure is a line in Dr. Whatif's Euclidean geometry.
Using this figure, or a close facsimile, make two constructions.
Construct m and explain your method. (The D- perpendicular bisector of AB is the D-line m so that the D-reflection of A is B.
Construct the D-circle with diameter AB and construct its D-center. Again, explain your reasoning.
Suppose that c and d are orthogonal circles. Explain, if true, why the inversion of d in c is d and the inversion of c in d is c.
Suppose c has center (0,0) and radius R. If circle d has equation x2 + y2 + Ax + By + c = 0, what is the equation of the inversion image of d?
Demonstrate that this is the equation of a circle (except in special cases – explain those cases).