Math: Geometry Problem 1023: Triangle, Contact, Gergonne, Product of three distances. Level: School, College.

In the figure below, P is a point on the incircle of triangle ABC and T_AT_BT_C is the contact triangle. If PA₁, PB₁, PC₁, PA₂, PB₂, and PC₂ are perpendicular to BC, AC, AB, T_BT_C, T_AT_C, and T_AT_B, respectively, Prove that
PA₁.PB₁.PC₁ = PA₂.PB₂.PC₂.
 
The contact triangle or intouch triangle or Gergonne triangle of a triangle ABC is defined by the 3 tangency points of the incircle on the 3 sides.