Question

$\large\tt\underline {Question:-}$

●ɪꜰ ᴛᴡᴏ ᴄɪʀᴄʟᴇꜱ ɪɴᴛᴇʀꜱᴇᴄᴛ ᴀᴛ ᴛᴡᴏ ᴘᴏɪɴᴛꜱ,ᴘʀᴏᴠᴇ ᴛʜᴀᴛ ᴛʜᴇɪʀ ᴄᴇɴᴛʀᴇ ʟɪᴇ ᴏɴ ᴛʜᴇ ᴘᴇʀᴘᴇɴᴅɪᴄᴜʟᴀʀ ʙɪꜱᴇᴄᴛᴏʀ ᴏꜰ ᴛʜᴇ ᴄᴏᴍᴍᴏɴ ᴄʜᴏʀᴅ?¿​

Answer 1

$\large\tt\underline {answer:-}$

Tᴡᴏ ᴄɪʀᴄʟᴇ ᴡɪᴛʜ ᴄᴇɴᴛʀᴇ O ᴀɴᴅ O′ɪɴᴛᴇʀsᴇᴄᴛ ᴀᴛ A ᴀɴᴅ B. AB ɪs ᴄᴏᴍᴍᴏɴ ᴄʜᴏʀᴅ ᴏғ ᴛᴡᴏ ᴄɪʀᴄʟᴇ OO′ ɪs ᴛʜᴇ ʟɪɴᴇ ᴊᴏɪɴɪɴɢ ᴄᴇɴᴛʀᴇ 

Lᴇᴛ OO′ ɪɴᴛᴇʀsᴇᴄᴛ AB ᴀᴛ P

Iɴ OAO ᴀɴᴅ OBO′ ᴡᴇ ʜᴀᴠᴇ 

OO′→ ᴄᴏᴍᴍᴏɴ

OA=OB→(ʀᴀᴅɪɪ ᴏғ ᴛʜᴇ sᴀᴍᴇ ᴄɪʀᴄʟᴇ)

O′A=O′B→(ʀᴀᴅɪɪ ᴏғ ᴛʜᴇ sᴀᴍᴇ ᴄɪʀᴄʟᴇ)

⇒ △OAO′≅△OBO′ {SSSᴄᴏɴɢᴜᴇɴᴄᴇ}

∠AOO′=∠BOO′ (CPCT)

ɪ.ᴇ., ∠AOP=∠BOP

Iɴ △AOP ᴀɴᴅ BOP ᴡᴇ ʜᴀᴠᴇ OP=OP ᴄᴏᴍᴍᴏɴ

∠AOP=∠BOP (ᴘʀᴏᴠᴇᴅ ᴀʙᴏᴠᴇ)

OA=OB (Rᴀᴅɪɪ ᴏғ ᴛʜᴇ sᴇᴍɪᴄɪʀᴄʟᴇ)

△APD=△BPD (SSSᴄᴏɴɢᴜᴇɴᴄᴇ)

AP=CP (CPCT)

ᴀɴᴅ ∠APO=∠BPO (CPCT)

Bᴜᴛ ∠APO+∠BPO=180

∴ ∠APO=90ᴏ

∴ AP=BP ᴀɴᴅ ∠APO=∠BPO=90ᴏ

∴ OO′ ɪs ᴘᴇʀᴘᴇɴᴅɪᴄᴜʟᴀʀ ʙɪsᴇᴄᴛᴏʀ ᴏғ AB

Answer 2

Tᴡᴏ ᴄɪʀᴄʟᴇ ᴡɪᴛʜ ᴄᴇɴᴛʀᴇ O ᴀɴᴅ O′ɪɴᴛᴇʀsᴇᴄᴛ ᴀᴛ A ᴀɴᴅ B. AB ɪs ᴄᴏᴍᴍᴏɴ ᴄʜᴏʀᴅ ᴏғ ᴛᴡᴏ ᴄɪʀᴄʟᴇ OO′ ɪs ᴛʜᴇ ʟɪɴᴇ ᴊᴏɪɴɪɴɢ ᴄᴇɴᴛʀᴇ

Lᴇᴛ OO′ ɪɴᴛᴇʀsᴇᴄᴛ AB ᴀᴛ P

Iɴ OAO ᴀɴᴅ OBO′ ᴡᴇ ʜᴀᴠᴇ

OO′→ ᴄᴏᴍᴍᴏɴ

OA=OB→(ʀᴀᴅɪɪ ᴏғ ᴛʜᴇ sᴀᴍᴇ ᴄɪʀᴄʟᴇ)

O′A=O′B→(ʀᴀᴅɪɪ ᴏғ ᴛʜᴇ sᴀᴍᴇ ᴄɪʀᴄʟᴇ)

⇒ △OAO′≅△OBO′ {SSSᴄᴏɴɢᴜᴇɴᴄᴇ}

∠AOO′=∠BOO′ (CPCT)

ɪ.ᴇ., ∠AOP=∠BOP

Iɴ △AOP ᴀɴᴅ BOP ᴡᴇ ʜᴀᴠᴇ OP=OP ᴄᴏᴍᴍᴏɴ

∠AOP=∠BOP (ᴘʀᴏᴠᴇᴅ ᴀʙᴏᴠᴇ)

OA=OB (Rᴀᴅɪɪ ᴏғ ᴛʜᴇ sᴇᴍɪᴄɪʀᴄʟᴇ)

△APD=△BPD (SSSᴄᴏɴɢᴜᴇɴᴄᴇ)

AP=CP (CPCT)

ᴀɴᴅ ∠APO=∠BPO (CPCT)

Bᴜᴛ ∠APO+∠BPO=180

∴ ∠APO=90ᴏ

∴ AP=BP ᴀɴᴅ ∠APO=∠BPO=90ᴏ

∴ OO′ ɪs ᴘᴇʀᴘᴇɴᴅɪᴄᴜʟᴀʀ ʙɪsᴇᴄᴛᴏʀ ᴏғ AB

❤❤MISSING MY OLD PAROMITA SIS ❤❤