Geometry Problem - Sum of distances

[franceboy]

Homework Statement

ABC is a triangle with I as the centre of the incircle and K as the centre of the circumcircle (I ≠ K). d(X) is the sum of the distances from a point X inside the triangle to the three sides. Verify if d(P)=d(Q) (P and Q are points inside the triangle), the lines PQ and IK intersect perpendicularly.

Homework Equations

d(X) = 3*r + f(x,φ) r is the radius of the incircle, x is the distance between X and I and φ is the angle between AB and IX.

The Attempt at a Solution

I have determined the complicated function f. Then I tried to find an connection between x,φ and y,ψ (d(X)=d(Y)) but the result was not very helpful. Another attempt was to try to find all points Y so that d(X)=d(Y). However I failed.

I do not search for a whole solution, but for good start or a theorem which could be useful.
Thank you very much :)

 

[Dr. Courtney]

A picture would help us see what you are trying to do here.

I expect the solution would involve the distance formula.