Page 3 - 微言“瓜豆原理”++浅谈“解题模型”

ZHONGXUESHUXUEZAZHI ᇏ࿐ඔ࿐ᄖᆽ 2019 ୍ֻ 8 ௹

A ູקׄđ ׄ B ᄝ϶ູࣥ r ֥ק ⊙O ഈᄎ׮đ ᄝᄎ׮ݖ ๎֥ԩ৘Ϸم．
ӱᇏđ Ќӻ ∠BAC = α̵ 0° ＜ α ＜ 180°̶ đ ౏ AC = k· 2．ၛ “ ҂эਈ ”“ ໊ර࿈ሇ” ్ູೆׄ．
AB̵ k ＞ 0̶ ҂э．ᄵׄ C ္ᄝ၂קჵഈᄎ׮． ̵ 1̶ ଁี 1 ֥ᆣૼ͇
2 ᆣૼଁี ೂ๭ 5đ ቔ AB ⊥ lđ ቔ
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гᆀՖೂ༯֥ਆ۱๯ࣥೆ൭࿙౰ࢳճҦ੻͇ AC đ ൐֤ ∠B AC = αđAC =
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1．ࠎႿ “ ໊ර࿈ሇ”đ ၛ “ਆׄקཌ ”、“೘ׄקჵ” k·AB đ ᆰཌ CC ࢌᆰཌ l Ⴟ
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ູ๬௥ׄ． ׄ Pđ ၞ ᆣđ △ACC 0 ∽
̵ 1̶ ଁี 1 ֥ᆣૼ͇ △ABB đ Ֆ طđ ∠AC C =
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๭ 5
ೂ๭ 3đ ֒ׄ B ᄎ׮֞ ∠AB B = 90°đ ࣉطđ β = α．
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B'đ ׄ C ෛᆭᄎ׮֞ C' ൈđ ৵ ཬࢲ ൙ൌഈđ ࢳथ׮ׄ໙ี֥၂۱ܱ࡯ׄ൞࿙
ࢤ BC、B'C' ࢌႿׄ Dđ ৵ࢤ ᅳ “ ҂эਈ”đ קׄ A ֞קᆰཌ l ֥ए৖———“ׄཌए
ADđ ᆰཌ CC' ࢌᆰཌ l Ⴟׄ P． ৖” ൞၂۱҂эਈđ ቔ్ູೆׄđ ڣၛ “໊ර࿈ሇэ
ၹ ູ ∠B'AC' = α = ߐ”đ ၛ “ קׄ·ק࢘” ູઋ࢖ׄᆣૼՖ׮ׄᄝ၂קᆰ
AC' AC ๭ 3 ཌഈᄎ׮．
∠BACđ = k = đ ෮ၛ
AB' AB ᆃဢ֥ࢳ൤҂ࣇࡥჿၞ׭đ ط౏္ಒקਔՖ׮ׄ
△AB'C' ∽ △ABCđ ෮ၛ ∠AB'C' = ∠ABCđ ౏ ∠AC'B' ֥݅ࠖ ——— ݖקׄ P̵ ᆣૼݖӱᇏၘടࠣఃೂޅಒ
= ∠ACBđ ෮ၛ AđBđB'đD ඹׄ܋ჵđ ౏ AđDđCđC' ඹׄ ק̶ ౏აקᆰཌ l ֥ࡃູ࢘קᆴ α．
܋ჵđ ෮ၛ ∠AB'B = ∠ADB = ∠AC'Cđ ෮ၛ AđB'đPđ ̵ 2̶ ଁี 2 ֥ᆣૼ͇
C' ඹׄ܋ჵđ ෮ၛ β = α．ࠧ೏ׄ BđC Ф౼קđ ᄵ CC' ა ᆣૼ͇ ೂ๭6đ ৵ࢤ AOđ ቔ
ᆰཌ l ֥ࡃູ࢘ αđ Ⴎ C' ֥಩ၩྟॖᆩđ ׄ C ᄝ၂קᆰ AO'đ ൐֤ ∠OAO' = αđ ౏ AO'
ཌഈᄎ׮． = k·AOđ ৵ࢤ BOđCO'đ ၞᆣđ
ཬࢲ ᄝ “ ཌྷර೘࢘ྙ” ֥ࠎԤഈđ ๙ݖ “ ඹׄ܋ CO'
△ACO' ∽ △ABOđ Ⴟ൞đ
ჵ”đ ၛ “ קׄ·ק࢘” ູઋ࢖ׄ൐ଁี֤ၛᆣૼ．༯໓ BO
ട֥ࠣ֞ཌྷර೘࢘ྙ֥ᆣૼაЧ২ᇏ֥ᆣૼ๝৘đ ҂ AC
= = kđ ՖطđCO' = k·rđ ࠧ
ᄜሕඍ． AB ๭ 6
̵ 2̶ ଁี 2 ֥ᆣૼ͇ ׄ C ᄝၛ O' ູჵྏđ ၛ k·r ູ
ೂ๭ 4đ ֒ׄ B ᄎ׮֞ ϶֥ࣥקჵഈᄎ׮．
B'đB″đ ׄ C ෛᆭᄎ׮֞ C'đC″ ཬࢲ קׄ A ֞קჵ֥ჵྏ O ֥ए৖———“ ׄྏ
ൈđ ඨՑ৵ࢤ BđB'đB″͈ CđC'đ ए৖” ൞၂۱ “ ҂эਈ”đ ၛՎ్ູೆׄđ ڣၛ “໊ර࿈
C″．ၞᆣ △ABB' ∽ △ACC'đ Ⴟ ሇэߐ”đ ၛ “ קׄ·קӉ”ູઋ࢖ׄᆣૼՖ׮ׄᄝ၂

CC' קჵഈᄎ׮．
൞đ = kđ ᄵ CC' = k·BB'đ
BB' หљֹđ ֒ α = 0° ࠇ 180° ൈđ ࢲં္Ӯ৫đ Վൈđ
ၞᆣđ △ABB″ ∽ △ACC″đ Ⴟ ๭ 4 ׄ C ᄝཌ؍ AB ࠇః࿼Ӊཌഈđ ೏ k = 1đ ᄵՎൈ֥эߐ
൞đ ∠AB″B = ∠AC″Cđ ၞᆣđ ູࠧ “ ࿈ሇэߐ”．๝ൈđ ၞᆩՖ׮֥ׄᄎ׮ਫ਼ӱ S' ࠧ
△AB'B″ ∽ △AC'C″đ Ⴟ ൞đ ∠AB″B' = ∠AC″C'đ ູᇶ׮֥ׄᄎ׮ਫ਼ӱ S ֥ k Пđ Ⴟ൞౰ࢳՖ׮֥ׄᄎ
∠AB″B － ∠AB″B' = ∠AC″C － ∠AC″C'đ ᄵ ∠BB″B' = ׮ਫ਼ӱൈđ ॉ੮ “ ᇶ׮ׄ”ࡗࢤ౰ࢳສສбॉ੮ “Ֆ׮
∠CC″C'đ ࠧ೏ׄ BđB' Ф౼קđ ᄵׄ CđC' ෛᆭФಒקđ
ׄ” ᆰࢤ౰ࢳ۷ູь০． طຓđ ᇭ؟აՎཌྷܱ֥౰ࢳ
Ⴛקཌ؍ BB' ֥ᅦ࢘ ∠BB″B' ҂эđ ᄵקཌ؍ CC' ֥ “ ֆ׮ׄ ——— ֆཌ؍” ֥ቋཬᆴ֥໙ีđ ӈӈॖၛ০
ᅦ࢘ ∠CC″C' ҂эđ ࠧׄ C″ ᄝ၂קჵഈᄎ׮． Ⴈ “ ܨ׸ჰ৘” ಒק “ ֆ׮ׄ” ֥݅ࠖđ طު০Ⴈ “ Լཌ
ཬࢲ ᄝ “ ໊ර࿈ሇ” ֥ࠎԤഈၛ “ קཌ·ק࢘”
؍ቋ؋” ࠇ “ ׄྏए৖ࡨ϶ࣥ” টಒק．
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ູઋ࢖֤ׄᆣ．ׄ B″ ބ C″ ٳљᄝਜ޶BB'đCC' ഈ֥౦ ֒ಖđ ࠎႿ “ ໊ර࿈ሇэߐ” ᆣૼਔ “ܨ׸ჰ৘”đ
ྙđ Վԩ҂ᄜሕඍ． ๝ൈđ ๝ဢ္ॖၛ൐Ⴈ “໊ර࿈ሇэߐ”ࢳճॖၛႨ
ഈඍ֥ᆣૼٚൔđ ؓႿ“ඹׄ܋ჵ”ބ“೘ׄק “ ܨ׸ჰ৘” ࢳճ֥໙ี．ೂݔ৘ࢳਔೂഈ֥ਆ۱ଁี
ჵ” ေ౰ࢠۚđ ࢝ഽބሧႪളၞႿ৘ࢳ．࡫Ⴟགྷྛ࢝ҋ ֥ᆣૼđ ପࣼ҂଴৘ࢳ “ܨ׸ჰ৘”֥൐Ⴈေژކс
ؓ “ ඹׄ܋ჵ”ടࠣ҂؟౏ေ౰ࢠ֮đ ֒࿙౰۷ູຈ ေ֥భิ่ࡱ．
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