Q:

Minimising the Sum of distance of two line formed by joining some point on a line to two fixed points

Consider the  xy coordinate system.
Let $l$ be the line $y=-x$.
Consider two points $A=(0,2)$ and $B=(2,3)$
The Question asks to find a point $C$ on the line $l$ such that $|AC|+|BC|$ is minimum.
The only method I got to tackle this problem is to use parametric form of a point on the line $l$, that is $(t,-t)$ and use calculus to minimize the distance as a function of variable $t$.
I tried to find the critical points using differentiation.
Unfortunately it gives me a $4$ degree equation with leading coefficient being $24$. Also since it has $4$ roots I an also getting duplicate roots
Any other better approach?

A:

If $A'=(-2,0)$ is the reflection of $A$ about line $l$, then $C$ is the intersection between $l$ and $A'B$. See here for a proof: https://www.cut-the-knot.org/Curriculum/Geometry/HeronsProblem.shtml.

