a) In a $ XYZ$ triangle, the incircle tangents the $ XY $ and $ XZ $ sides at the $ T $ and $ W $ points, respectively. Prove that: $$ XT = XW = \frac {XY + XZ-YZ} {2} $$Let $ ABC $ be a triangle and $ D $ is the foot of the relative height next to $ A. $ Are $ I $ and $ J $ the incentives from triangle $ ABD $ and $ ACD $, respectively. The circles of $ ABD $ and $ ACD $ tangency $ AD $ at points $ M $ and $ N $, respectively. Let $ P $ be the tangency point of the $ BC $ circle with the $ AB$ side. The center circle $ A $ and radius $ AP $ intersect the height $ D $ at $ K. $ b) Show that the triangles $ IMK $ and $ KNJ $ are congruent c) Show that the $ IDJK $ quad is inscritibed