(ma+nb):b::(mc+nd):d, prove a,b,c,d are in proportion by using componendo and dividendo

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Solution

ma+nb = mc+nd
b d

d(ma+nb)=b(mc+nd)

dma+dnb=bmc+bnd

dma-bmc=bnd-dnb

m(da-bc)=n(bd-db)

m(da-bc)=n(0)

m(da-bc)=0

da-bc=0
m

da-bc=0

da=bc

a = c
b d

=> thus, a,b,c and d are in proportion.
hence proved

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