Maximum distance of $P$ from side $BC$

Question

Let $P$ be a point which doesnot lie outside the triangle $ABC$,$A(3,2),B(0,0),C(0,4)$ and satisfies $$d(P(A))\geq \text{ }max [d(P,B),d(P,C)] $$then the question is to find out the maximum distance of $P$ from side $BC$ where $d(P,A)$ gives the distance between $P$ and $A$

I tried to solve this problem using coordinate geometry.Taking the point $P$ as $(x,y)$ I tried to solve for the conditions given but it didnot helped me figure out a definite locus of the point $P$ which could help me to solve me to solve this problem.Any ideas?Thanks.

Answer 1

First, note that by definition, the distance from any point $P$ on the perpendicular bisector of a segment $AB$ to either $A$ or $B$ is the same. Once the point is on one side of the line, the corresponding point is the closest one.

So your conditions on $P$ just mean that the point lies in the triangle $ABC$, on one side of the bisector of $AB$ and on one side of the bisector of $AC$.

Try to continue from there!