Question

In ΔABC, the bisector of ∠A intersects BC in D and the bisector of ∠ADC intersects AC in E. Prove that AB X AD X EC = AC X BD X AE.

Answer 1

Given : in ∆ABC, the bisector of ∠A intersects BC in D and the bisector of ∠ADC intersects AC in E as shown figure.

To prove : AB × AD × EC = AC × BD × AE

proof : in ∆ABC, the bisector of ∠A intersects BC in D.
according to angle bisector theorem,
BD/DC = AB/AC .........(i)

in ∆ADC, the bisector of ∠ADC intersects AC in E.
according to angle bisector theorem,
AE/EC = AD/DC .........(ii)

multiplying eqs. (i) and (ii),
(BD × AE)/(DC × EC) = (AB × AD)/(AC × DC)
(BD × AE)/EC = (AB × AD)/AC
BD × AE × AC = AB × AD × EC
AC × BD × AE = AB × AD × EC
hence proved.