Question

two chords AB and CD intersect anywhere inside the circle at an angle of 90 degrees. prove that arc(CB)+ arc(AD)= arc(AC) + arc(BD)

Answer 1

Heya,

Friend,

Hope this will help you!!

Answer 2

let the point of intersection of AB and CD be P.
join ac,cb,bd,ad
let AC be a
BC be b
BD be c
and AD be D
AP^2+PC^2=a^2____(1)
PC^2+PB^2=b^2____(2)
SUBTRACT (1) AND (2)
AP^2-PB^2=a^2-b^2____(3)
PD^2+PB^2=c^2____(4)
PD^2+PA^2=d^2____(5)
SUBTRACT(4) AND (5)
AP^2-PB^2=d^2-c^2_____(6)
EQUATE (3) AND (6)
a^2-b^2=d^2-c^2
a^2+c^2=b^2+d^2
NOW,
ACCORDING TO TOULMIN'S THEOREM
a*c=b*d
2*a*c=2*b*d
ADD THIS IN ABOVE EQUATION
a^2+c^2+2*a*c=b^2+d^2+2*b*d
(a+c)^2=(b+d)^2
a+c=b+d
SO ACCORDING TO OPPOSITE SEGMENT THEOREM
PROVED