Question

in a triangle ABC, if D is a point on BC such that BD/DC=AB/AC,prove that AD is the bisector of <A​

Answer 1

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In a triangle ABC, if D is a point on BC such that BD/DC=AB/AC,prove that AD is the bisector of <A.

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ʙᴅ/ᴅᴄ = ᴀʙ/ᴀᴄ

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< ʙᴀᴅ = < ᴄᴀᴅ

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ᴇxᴛᴇɴᴅ ʙᴀ ᴛᴏ ᴇ, ꜱᴜᴄʜ ᴛʜᴀᴛ ᴀᴇ = ᴀᴄ .

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ꜱɪɴᴄᴇ ᴀᴇ = ᴀᴄ

ʜᴇɴᴄᴇ < ᴀᴇᴄ = < ᴀᴄᴇ

ꜱɪɴᴄᴇ, ʙᴅ/ ᴄᴅ = ᴀʙ/ᴀᴄ ( ɢɪᴠᴇɴ)

ʙᴜᴛ , ᴀᴄ = ᴀᴇ

=> ʙᴅ/ᴄᴅ= ᴀʙ/ ᴀᴇ

=> ᴀᴅ // ᴇᴄ ______( \\ means parallel )

( ᴀꜱ ɪɴᴛᴇʀᴄᴇᴘᴛꜱ ᴏɴ ᴛʀᴀɴꜱᴠᴇʀꜱᴀʟꜱ ʙᴄ & ʙᴇ ᴀʀᴇ ᴘʀᴏᴘᴏʀᴛɪᴏɴᴀʟ)

ꜱᴏ, < ᴄᴇᴀ = < ᴅᴀʙ ( ᴄᴏɴꜱᴇᴄᴜᴛɪᴠᴇ ɪɴᴛᴇʀɪᴏʀ ᴀɴɢʟᴇꜱ ꜰᴏʀᴍᴇᴅ ʙʏ // ʟɪɴᴇꜱ) ______(1)

< ᴇᴄᴀ = < ᴄᴀᴅ (ᴀʟᴛᴇʀɴᴀᴛᴇ ɪɴᴛᴇʀɪᴏʀ ᴀɴɢʟᴇꜱ ᴀᴅ// ᴇᴄ )__

_____________________ (2)

ʙᴜᴛ < ᴄᴇᴀ = < ᴇᴄᴀ

ᴛʜᴇʀᴇꜰᴏʀᴇ, < ᴅᴀʙ= < ᴄᴀᴅ

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