Three circles with radii 3cm, 4cm and 5cm touch each other externally. If A is the point of intersection of tangents to these circles at their points of contact, then the distance of A from the points of contact is

It is clear that A is the centre of the incircle of the triangle PQR formed by the lines joining the centres of the given circles. We have PQ=5+4=9, QR=4+3=7, RP=3+5=8,  distance of A from the points of contact is the radius r of the incircle of triangle PQR.From trigonometry r=Δ/s  where r=Δ/s is the area and 2s  is the perimeter of the triangle PQRΔ=12(12−7)(12−8)(12−9)=12×5×4×3                                                   =125s=(1/2)(9+7+8)=12,Hence r=5