Point P with two cevians bounding a quadrilateral and a triangle having congruent incircles

1) Let A' be the midpoint between B and C.
2) Let D be the contact point between BC and the incircle.
3) The lines A'I (I incenter) and AD intersect at Q.
4) The perpendicular from I to A'I cuts the circle with diameter A'Q at S.
5) The circle centered at A' and going through S cuts a'Q at the requested point P.