#a) x*15=5*(5+13)#
#=> x=(5*18)/15=6#
#b) x*20=16^2#
#=> x=16^2/20=256/20=64/5=12.8#
1) proof of the two-secant theorem:
Let #angleSOQ=x#,
#PQRS=#cyclic quadrilateral, => #anglePSR=anglePQR=y#
#=> DeltaORQ and DeltaOPS# are similar,
#=> (OR)/(OQ)=(OP)/(OS)#
#=> c/(a+b)=a/(c+d)#
#=> a(a+b)=c(c+d)# proved.
2) proof of the tangent-secant theorem:
See Fig.1.
Let #T# be the center of the circle.
#angle OPT=90^@#
Let #angle OPR=y, => angle TPR=90-y=angleTRP#
#=>anglePTR=2y, => anglePSR=y#
#=> angleOPR=anglePSR#, a well-known property of tangents
See Fig 2.
Let #angleOPR=y, =>anglePSR=y#
Let #anglePOS=x#
#=> Delta OPR and DeltaOSP# are similar.
#=> (OR)/(OP)=(OP)/(OS)#
#=> c/a=a/(c+d)#
#a^2=c(c+d)# proved.
