State whether each of the following statements is true or false:
$(1)$ There are three circles passing through three given non-collinear points.
$(2)$ Infinitely many circles pass through two given points.
$(1)$ There are three circles passing through three given non-collinear points.
$(2)$ Infinitely many circles pass through two given points.
(A) $(1)$ False. There is only one unique circle that passes through three given non-collinear points.
$(2)$ True. Infinitely many circles can be drawn passing through two given points,as their centers can lie anywhere on the perpendicular bisector of the line segment joining the two points.
$(2)$ True. Infinitely many circles can be drawn passing through two given points,as their centers can lie anywhere on the perpendicular bisector of the line segment joining the two points.
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