In $$\triangle ABC, \angle B = 90^\circ$$, If the points D and E are on the side BC such that BD = DE = EC then which of the following is true?

Let the BD, BE and BC be x, 2x and 3x respectively.

In $$\triangle$$ ABD,

$$(AD)^2 = (AB)^2 + (BD)^2$$

$$(AD)^2 = (AB)^2 + (x)^2$$ ----(1)

In $$\triangle$$ ABE,

$$(AE)^2 = (AB)^2 + (BE)^2$$

$$(AE)^2 = (AB)^2 + (2x)^2$$

$$(AE)^2 = (AB)^2 + 4x^2$$ ---(2)

In $$\triangle$$ ABC,

$$(AC)^2 = (AB)^2 + (BC)^2$$

$$(AC)^2 = (AB)^2 + (3x)^2$$

$$(AC)^2 = (AB)^2 + 9x^2$$ ---(3)

Eq(1) multiply by 4,

$$4AD^2 = 4AB^2 + 4x^2$$ ---(4)

Eq(4) - (2),

$$4AD^2 - AE^2 = 3AB^2$$ ---(5)

Eq(1) multiply by 9,

$$9AD^2 = 9AB^2 + 9x^2$$ ---(6)

Eq(6) - (3),

$$9AD^2 - AC^2 = 8AB^2$$ ---(7)

Eq(5) multiply by 8 and Eq(7) multiply by 3,

$$32AD^2 - 8AE^2 = 24AB^2$$ ---(8)

$$27AD^2 - 3AC^2 = 24AB^2$$ ---(9)

From eq(8) and (9),

$$32AD^2 - 8AE^2 = 27AD^2 - 3AC^2$$

$$ 8AE^2 = 5AD^2 + 3AC^2$$