Question

in an isosceles triangle ABC ,AB=AC and D is a point on BC produced. prove that
AD×AD = AC×AC + BD×CD​

Answer 1

$\huge \underline \mathfrak \orange{Hello\: mate!!}$

try to solve with pythagoras theorem

Answer 2

ANSWER

Let AX be a line perpendicular to BC and as perpendicular from a vertex common to equal side is also a median therefore, BX=CX=

2

1

(BC)

Now, By Pythagoras theorem in ΔADX and ΔACX,

AD

2

=AX

2

+XD

2

and AC

2

=AX

2

+CX

2

⇒AD

2

−AC

2

=XD

2

−CX

2

=(BD−BX)

2

−CX

2

=BD

2

+BX

2

−2BD.BX−CX

2

(BX=CX)

=BD(BD−2BX)

=BD(BD−BC)

⇒AD

2

−AC

2

=BD⋅CD

∴AD

2

=AC

2

+BD⋅CD