If the altitude from one vertex of a triangle bisects the opposite side, then the triangle is isosceles. GIVEN : A `A B C` such that the altitude `A D` from `A` on the opposite side `B C` bisects `B C` i.e., `B D=D Cdot` TO PROVE : `A B=A C` i.e. the triangle `A B C` is isosceles.

Explanation

`BO=DC`

`In /_ABD and /_ACD`

`AD=AD`(common)

`/_ADB=/_ADC`(each 90)

`BD=CD`(given)

`/_ABD cong /_ACD`(SAS)

`AB=AC`

ABC is isosceles.