Question

2. In the figure, two circles intersect each
other at points A and B. C is a point on
the smaller circle. Secant CA and secant
CB of the smaller circle intersects the
bigger circle at points D and E
respectively then prove : seg DE || line
CF.​

Answer 1

Answer:

construction: segment AB

proof:

angle E+angle DAB=180°.....opposite angle of a cyclic quadrilateral..i

angle DAB= angle ACB+angle CBA. .....remote interior angle...ii

angle CBA=1÷2×arcAFC..inscribed angle theorem...iii

angleACF=1÷2×arcAFC....theorum of the angle between tangent and decent..iv

from..iii and...iv,

angle CBA= angleACF...v

hence, angle DAB= angleACB +angle ACF...from ii and v

that is, angle DAB=angle BCF....(angle BCA+ angleAFC=angleBCF).....vi

hence from i and vi,

angle E + angle BCF=180°

hence according to the reverse of corresponding angle theorum,

seg DE || line CF.

hence proved.