Question

prove that the centre of a circle touching two intersecting limes lies on the angle bisector of the lines???

Answer 1

i) Let the two lines l₁ and l₂ intersect at point P. And the circle touches the two lines at A and B respectively. Join center O of the circle with A and B respectively. Also join OP.  

ii) At the point of contact radius and tangent are perpendicular.  

So, <OAP = <OBP = 90 deg  

OP = OP [Common]  

OA = OB [Radii of the same circle]  

Hence ΔOAP ≅ ΔOBP [RHS Congruence axiom]  

So <OPA = <OPB [Corresponding parts of congruence triangles are equal]  

Hence OP is the bisector of angle APB.  

Thus it is proved that O lies on the bisector of the angle formed by the intersecting lines.