Draw a general* triangle ABC and construct with straightedge and compass the perpendicular bisectors of the sides.
(a) Prove that the 3 perpendicular bisectors are concurrent at a point O for any triangle ABC.
(b) Construct the circle with center O through A.
(c) Explain why the circle with center O through A also passes through B and through C for any triangle ABC.
*General is a somewhat ill-defined term, but referring to an example, it means not special, so it should not be isosceles, equilateral, having a right angle or any other "special" angle such as 60 degrees. It means what one might call a "random" triangle if one could give a good definition of random.
Suppose that ABC and DEF are triangles, with AB = DE, AC = DF and angle B = angle E. This condition is called SSA (or you can reverse the order of letters if you like!).
Construct an example that shows that SSA is not a valid criterion for triangle congruence. Specifically, find two lengths AB and AC, and an angle B and then construct two triangles A'B'C' and A"B"C" that satisfy the SSA criterioin but are not congruent.