Question

Given a triangle with unequal sides, if P is the set of all points which are equidistant from B and C, and Q is the set of all points which are equidistant from sides AB and AC, then what is the P intersection with Q equal to?

Answer 1

GIVEN : A Scalene teiangle $ABC$.

$P$ is a set of points which are equidistant from point $B$ & point $C$

=> Every point, belonging to Set P lies on the perpendicular bisector of $BC$. This perpendicular bisector can be extended up to infinity.

=> cardinal ( Set $P$) = infinite elements

Now, Given that set $Q$ , contains elements , equidiastant from side $AB$ & $AC$. Hence locus of Q will be the angle bisector of < A. ( as we know that , incentre $I$ is equidistant from all $3$ sides $AB,AC$, & $BC$)

So, every point of angle bisector of $\angle A$ , will be equidistant from $AB$, & $AC$.

So, cardinal( Set $Q$) = infinite elements

TO FIND: $PnQ$ =? Or, $P$ intersection $Q=?$ ie, the element common to Set $P$ & Set $Q$ both=?

So, common element(point) is the point of intersection of perpendicular bisector of $BC$ & angle bisector of $\angle A$

=>$PnQ=$ the point of intersection of perpendicular bisector of $BC$ & angle bisector of $\angle$A$. And since triangle is scalene triangle,

Hence , cardinal $\left(PnQ\right)=1$