Write 'True' or 'False' and give reasons for your answer.
If a number of circles touch a given line segment $PQ$ at a point $A$,then their centres lie on the perpendicular bisector of $PQ$.
If a number of circles touch a given line segment $PQ$ at a point $A$,then their centres lie on the perpendicular bisector of $PQ$.
(B) False.
Let $PQ$ be a line segment and $A$ be a point on it. If several circles touch the line segment $PQ$ at point $A$,then the radius of each circle at the point of contact $A$ is perpendicular to the line segment $PQ$.
Let the centres of these circles be $C_1, C_2, C_3, \dots$. Since each circle touches the line segment $PQ$ at point $A$,the line segment connecting the centre of each circle to the point $A$ (i.e.,$C_1A, C_2A, C_3A, \dots$) must be perpendicular to $PQ$ at point $A$.
This means that all the centres $C_1, C_2, C_3, \dots$ lie on a line that is perpendicular to $PQ$ at point $A$. However,for these centres to lie on the perpendicular bisector of $PQ$,the point $A$ must be the midpoint of $PQ$. Since the problem statement does not specify that $A$ is the midpoint of $PQ$,the centres lie on a line perpendicular to $PQ$ at $A$,but not necessarily on the perpendicular bisector of $PQ$.
Let $PQ$ be a line segment and $A$ be a point on it. If several circles touch the line segment $PQ$ at point $A$,then the radius of each circle at the point of contact $A$ is perpendicular to the line segment $PQ$.
Let the centres of these circles be $C_1, C_2, C_3, \dots$. Since each circle touches the line segment $PQ$ at point $A$,the line segment connecting the centre of each circle to the point $A$ (i.e.,$C_1A, C_2A, C_3A, \dots$) must be perpendicular to $PQ$ at point $A$.
This means that all the centres $C_1, C_2, C_3, \dots$ lie on a line that is perpendicular to $PQ$ at point $A$. However,for these centres to lie on the perpendicular bisector of $PQ$,the point $A$ must be the midpoint of $PQ$. Since the problem statement does not specify that $A$ is the midpoint of $PQ$,the centres lie on a line perpendicular to $PQ$ at $A$,but not necessarily on the perpendicular bisector of $PQ$.
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