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ᴛʜᴇ ᴘᴇʀᴘᴇɴᴅɪᴄᴜʟᴀʀ ғʀᴏᴍ 'ᴀ' ᴏɴ ᴛʜᴇ sɪᴅᴇ ʙᴄ ᴏғ  ∆ᴀʙᴄ ɪɴᴛᴇʀsᴇᴄᴛs ʙᴄ ᴀᴛ ᴅ sᴜᴄʜ ᴛʜᴀᴛ ʙᴅ:ᴄᴅ = 3:2. ᴘʀᴏᴠᴇ ᴛʜᴀᴛ 

${AB}^{2} - {AC}^{2} = \frac{1}{5} \: {BC}^{2}$
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Answer 1

let BD =3x

DC = 2x

in ABD.

AD^2 = AB^2 - BD^2 = AB^2 - 9x^2

In ADC

AD^2 = AC^2 - DC^2 = AC^2 - 4 X^2

COMPARE

AB^2 - 9x^2 = AC^2 - 4x^2

AB^2 - AC^2 = -4x^2 + 9x^2 = 5x^2

BC= 3x+ 2x = 5x

BC^2 = 25 x^2

1/5 . BC^2 = 5x^2

So AB^2 - AC^2 = 1/5. BC^2

Answer 2

See solution in image .....

$\huge\underline\mathfrak\green{Hope\:it\:Helps\:You}$