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

Question

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

Infinity Learn · Accepted Answer

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 $$r=5$$

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

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