If a:b=c:d then √(a^2+c^2):√(b^2+d^2)=(pa+qc):(pb+qd)
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Let a:b=c:d=e:f
∴a=bk
∴c=dk
∴e=fk
Now
pb+qd+rf
pa+qc+re
=
pb+qd+rf
pbk+qdk+rfk
=
(pb+qd+rf)
k(pb+qd+rf)
∴(pa+qc+re):(pb+qd+rf)=k:1=a:b=c:d=e:f
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