Given
xcos alpha+ysin alpha-k=0....[1]
xcos beta+ysin beta-k=0....[2]
To show that x/cos ((alpha+beta)/2)=y/sin ((alpha+beta)/2)=k/cos ((alpha-beta)/2)
On cross multiplication of [1] an[2] we get
x/(sinbeta -sin alpha)=y/(cosbeta-cosalpha)=k/(sinbetacosalpha-cosbetasinalpha)
=>x/(-2cos((alpha +beta)/2)sin((alpha-beta)/2))=y/(-2sin((alpha+beta)/2)sin((alpha-beta)/2))=k/(-sin(alpha-beta))
=>x/(-2cos((alpha +beta)/2)sin((alpha-beta)/2))=y/(-2sin((alpha+beta)/2)sin((alpha-beta)/2))=k/(-2sin((alpha-beta)/2)cos((alpha-beta)/2))
=>x/(cos((alpha +beta)/2))=y/(sin((alpha+beta)/2))=k/(cos((alpha-beta)/2))