Angle Bisector of an Inscribed Triangle

Geometry Level 3

In the above figure $\triangle$ ABC is inscribed in the circle and DF is the angle bisector of $\angle$ ACE. If AD = 8 and AC = 6, find BD

Note : Figure is not drawn to scale

1 solution

Suresh Bala
Sep 1, 2014

Let $\angle$ BCD = x

$\angle$ FCE = x (vertical angles are equal)

$\angle$ BAD = x (angles by a chord in the same segment are equal)

$\angle$ ACF = x(DF is angle bisector )

Now, in ACD, $\angle$ ACD = 180-x

$\angle$ ABD = x(angles in a cyclic quadrilateral)

Hence, in $\triangle$ ABD, $\angle$ A = $\angle$ B i.e. AD = BD = 8

I think you can drop the assumption about length of $AC$ .

pierantonio legovini - 6 years, 1 month ago

True but sometimes extra data is dangerous and that's the purpose of AC here :)

Suresh Bala - 6 years ago