Question

Let ABC be a right-angled triangle with ∠B = 90°. Let BD be the altitude from B on to AC. Let P, Q and I be the in centres of triangles ABD, CBD and ABC respectively. Show that the circumcentre of the triangle PIQ lies on the hypotenuse AC.

Answer 1

Now if draw a circle with centre as R and radius as RP(=RQ) then ∠ made by PQ at centre = π/2

=> ∠ made by PQ at major segment = π/4 =>  ∠ made by PQ at minor segment = 3π/4 = ∠PIQ

=> hence I lies on this circle with centre at R.

And hence clearly circumcentre of ΔPIQ (i.e. R) lies on AC.