`A B C` is a triangle and `D` is the mid-point of `B C` . The perpendiculars from `D` to `A B` and `A C` are equal. Prove that the triangle is isosceles.
.Given : D is the mid-point of BC and PD=DQ.
To prove : `triangle ABC` is isosceles.
Proof
Let DE and DF be the perpendiculars from D on AB and AC respectively.
In `triangles BDE and CDF`
DE=DF ( given )
`angle BED = angle CFD = 90^*`
BD=DC ( Dis the mid-point of BC )
`therefore triangle BDE~=triangle CDF ` ( By RHS )
`angle B = angle C` ( by CPCT )
AC=AB ( Sides opposites to equal angles are equal )
`therefore triangle ABC` is isosceles.

Explanation

Given : D is the mid-point of BC and PD=DQ.
To prove : `triangle ABC` is isosceles.
Proof
Let DE and DF be the perpendiculars from D on AB and AC respectively.
In `triangles BDE and CDF`
DE=DF ( given )
`angle BED = angle CFD = 90^@`
`BD=DC` ( Dis the mid-point of BC )
`therefore triangle BDE~=triangle CDF ` ( By RHS )
`angle B = angle C` ( by CPCT )
`AC=AB` ( Sides opposites to equal angles are equal )
`therefore triangle ABC` is isosceles.