A point `D` is on the side `B C` of an equilateral triangle `A B C` such that `D C=1/4\ B C` . Prove that `A D^2=13\ C D^2` .

In an equilateral triangle A B C if A D_|_B C , then

In an equilateral triangle A B C the side B C is trisected at D . Prove that 9\ A D^2=7\ A B^2

In an equilateral triangle ABC, D is a point on side BC such that B D=1/3B C . Prove that 9A D^2=7A B^2 .

D is a point on the side BC of a triangle ABC such that /_A D C=/_B A C . Show that C A^2=C B.C D .

In Figure, if A B C is an equilateral triangle. Find /_B D C\ a n d\ /_B E C

D and E are points on equal sides A B and A C of an isosceles triangle A B C such that A D=A E . Prove that B ,C ,D ,E are concylic.

If D and E are points on sides A B and A C respectively of a triangle A B C such that D E || B C and B D=C E . Prove that A B C is isosceles.

The perpendicular A D on the base B C of a triangle A B C intersects B C at D so that D B=3C D . Prove that 2\ A B^2=2\ A C^2+B C^2 .

In an equilateral A B C , A D_|_B C , prove that A D^2=3\ B D^2

If D is any point on the base B C produced, of an isosceles triangle A B C , prove that A D > A B .