Normal as the Angle Bisector

By the definition of an ellipse, any point $$P$$ on the curve satisfies $$PF_1 + PF_2 = 2a$$. For any point $$Q$$ lying strictly on the tangent line $$L$$ at point $$P$$ (excluding $$P$$ itself), it falls outside the ellipse, meaning $$QF_1 + QF_2 > 2a$$.
 point $$P$$ is the unique path-minimizing point along the line $$L$$ for distances connecting $$F_1$$ and $$F_2$$.
According to Fermat's Principle of Least Time, light reflecting off a surface minimizes travel distance, meaning the angle of incidence equals the angle of reflection relative to the tangent line:
$$\angle (F_1P \text{ and } L) = \angle (F_2P \text{ and } L)$$

 Because the tangent line acts as the perfect external bisector of the focal lines, the normal line—which is defined as being strictly perpendicular to the tangent line—must act as the internal angle bisector of $$\angle F_1PF_2$$.