Question

AB x BC = BP x CA
If the above statement is true then mention answer as 1, else mention 0 if false

Answer 1

Given: $\angle ABC=2\angle ACB$, BP bisects $\angle ABC$
$\angle ABC=2\angle ACB$
$\frac{1}{2}\angle ABC=\angle ACB$
$\angle ABP=\angle ACB$ (AP bisects $\angle ACB$) (I)
Now, In $△APB$ and $△ABC$
$\angle ABP=\angle ACB$ (From I)
$\angle BAP=\angle BAC$ (Common)
$\angle APB=\angle ACB$ (third angle)
thus, $△APB\sim △ABC$
Hence, $\frac{AB}{AC}=\frac{BP}{BC}$...(Corresponding sides)
$AB\times BC=BP\times AC$