CD and GH are respectively the bisectorsof ACB and EGF in such a way that D and H lieon sides AB and FE of Δ ABC and Δ EFGrespectively. If Δ ABC ~ Δ FEG, show that:

(i)


(ii) Δ DCB ~ Δ HGE


(iii) Δ DCA ~ Δ HGF


It is given that ΔABC ΔFEG


A = F, B = E, and ACB = FGE


ACB = FGE


ACD = FGH (Angle bisector)


And, DCB = HGE (Angle bisector)


(i) In ΔACD and ΔFGH,


A = F (Proved above)


ACD = FGH (Proved above)


ΔACD ΔFGH (By AA similarity criterion)


(Corresponding sides of similar triangles are proportional)


(ii) In triangle DCB and HGE


DCB = HGE (Proved above)


B = E (Proved above)


Therefore,


ΔDCB ΔHGE (By AA similarity criterion)


(iii) In ΔDCA and ΔHGF,


ACD = FGH (Proved above)


A = F (Proved above)


ΔDCA ΔHGF (By AA similarity)


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