Tus thawj kos npe rau ntawm koob pheej ntawm lawv ntawm cov vajvoos peb sab. Qhov thib ob thiab thib peb cim ntawm koob pheej ntawm lawv ntawm cov vajvoos peb sab

Ntawm cov loj loj tus naj npawb ntawm cov uas, uas yog yeej tseem zoo uas tsis yog-sib tshuam kaw polygonal kab, ib tug daim duab peb sab - yog ib daim duab nrog rau cov tsawg kawg yog tus naj npawb ntawm cov ces kaum. Nyob rau hauv lwm yam lus, nws yog ib tug yooj yim polygon. Tab sis, txawm nws simplicity, qhov no daim duab conceals ib tug ntau ntawm mysteries thiab nthuav discoveries, uas tseem ceeb rau ib tug tshwj xeeb ceg ntawm kev kawm txog zauv - geometry. Qhov no kev qhuab qhia nyob rau hauv cov tsev kawm ntawv pib qhia rau cov qib xya, thiab "Daim duab peb sab" lub ntsiab yog muab tshwj xeeb mloog. Cov me nyuam tsis tau tsuas yog kawm txog cov kev cai ntawm daim duab nws tus kheej, tab sis kuj los sib piv lawv txoj kev kawm ntawv 1, 2 thiab 3, ib tug kos npe rau ntawm koob pheej ntawm lawv ntawm cov vajvoos peb sab.

Tus thawj acquaintance

Ib qho ntawm thawj cov kev cai, yog me ntsis txog cov cov menyuam kawm ntawv, nws mus ib yam dab tsi zoo li no: lub sum ntawm cov ces kaum ntawm ib tug daim duab peb sab sib npaug 180 degrees. Yuav kom paub meej tias qhov no, nws suffices siv cov protractor los ntsuas txhua lub vertices thiab ntxiv mus rau tag nrho cov uas ua qhov tseem ceeb. Raws li, thaum ob tug paub qhov tseem ceeb yooj yim txiav txim rau peb. Piv txwv li: Nyob rau hauv ib lub ces kaum ntawm lub daim duab peb sab yog 70 °, thiab lwm yam yog - 85 °, yog dab tsi qhov luaj li cas ntawm peb lub?

180 - 85 - 70 = 25.

Teb: 25 °.

Paub tab yuav ua tau nyuab dua, yog tias tsuas yog ib qho qee lub nqi thiab ib tug thib ob nqi txog hais tias tsuas yog nyob rau ntau npaum li cas los yog yuav ua li cas muaj ntau zaus nws yog ntau dua los yog tsawg dua.

Nyob rau hauv daim duab peb sab los mus txiav txim rau ib los yog lwm ntawm nws cov nta tshwj xeeb ntawm txoj kab, txhua tus uas yuav tau ris nws muaj nws tus kheej lub npe:

• qhov siab - lub perpendicular txoj kab los ntawm cov kab los sib ntsib rau qhov opposite sab;
• tag nrho peb heights, ua nyob rau tib lub sij hawm, nyob rau hauv qhov chaw ntawm lub daim duab tshuam, txoj kev orthocenter, uas, nyob ntawm seb lub hom daim duab peb sab yuav ua tau ob qho tib si sab hauv thiab sab nraud;
• Theem nrab - lub kab txuas rau sab saum toj mus rau nruab nrab ntawm qhov opposite sab;
• yog tus taw tes ntawm kev tshuam ntawm lub medians ntawm nws heev, yog nyob rau hauv lub qhov zoo lawm;
• bisector - kab khiav los ntawm cov saum toj kawg nkaus rau lub point ntawm kev tshuam uas muaj cov lus rov sab, lub taw tes rau ntawm kev tshuam ntawm lub peb bisectors yog qhov chaw ntawm lub inscribed voj voog.

Tej yam yooj yim txog tej kev tseeb txog cov vajvoos peb sab

Cov vajvoos peb sab, raws li, tseeb, thiab tag nrho cov nuj nqis muaj lawv tus kheej cov yam ntxwv thiab cov khoom. Raws li twb tau hais, qhov no daim duab yog ib tug yooj yim polygon, tab sis, nrog rau nws tus kheej cov yam ntxwv nta:

• tiv thaiv cov ntev heev-sab lub yeej ib txwm dag nrog ib tug loj qhov ntau, thiab vice versa;
• tiv thaiv cov sib npaug zos sab yog muaj sib npaug ces kaum, piv txwv li - ib tug ceskaum daim duab peb sab;
• lub sum ntawm lub sab hauv ces kaum yog ib txwm sib npaug zos rau 180 °, uas twb tau pom hais tias nyob rau hauv ib qho piv txwv;
• extending ntawm ib sab ntawm lub duab peb sab yog tsim tshaj lub txheej lub uas yuav yeej ib txwm ua ncaj rau lub sum ntawm cov ces kaum, nws twb tsis nyob ib sab;
• ib yam ntawm cov ob tog yog yeej ib txwm tsawg tshaj li tus sum ntawm cov lwm yam ob sab, tiam sis feem ntau ntawm lawv cov sib txawv.

hom ntawm cov vajvoos peb sab

Nrhiav rau cov theem tom ntej yog kom paub tias cov pab pawg neeg uas tus hais daim duab peb sab. Teej tug mus rau ib tug yam nyob rau qhov tseem ceeb ntawm cov ces kaum ntawm ib tug daim duab peb sab.

• Ceskaum - nrog ob tug sib npaug zos tog neeg uas hu ua sab, peb nyob rau hauv cov ntaub ntawv no ua raws li lub hauv paus nrhiav. Lub ces kaum ntawm lub hauv paus ntawm lub daim duab peb sab yog cov tib yam thiab theem nrab los ntawm cov saum toj kawg nkaus, yog lub bisector thiab qhov siab.
• Yog, los yog ib tug equilateral triangle - yog ib tug nyob rau hauv tag nrho cov uas nws sab yog muaj sib npaug.
• Lub ib qho ntawm nws cov ces kaum yog 90 °. Nyob rau hauv cov ntaub ntawv no, lub sab opposite no lub yog hu ua tus hypotenuse, thiab lwm yam ob - ob txhais ceg.
• Mob daim duab peb sab - tag nrho cov ces kaum tsawg dua 90 °.
• Obtuse - ib qho ntawm cov ces kaum ntau dua 90 °.

Koob pheej ntawm lawv thiab kev zoo sib thooj ntawm cov vajvoos peb sab

Nyob rau hauv tus txheej txheem ntawm txoj kev kawm yog tsis tsuas suav nyias coj zoo, tab sis kuj los sib piv ob tug cov vajvoos peb sab. Thiab qhov no seemingly yooj yim ntsiab muaj ib tug ntau ntawm cov kev cai thiab theorems uas yuav tsum tau muaj pov thawj hais tias qhov suav hais tias daim duab - sib npaug zos cov vajvoos peb sab. Tej yam tshwm sim ntawm lub duab peb ceg muaj ib tug txhais ntawm koob pheej ntawm lawv: lub duab peb ceg yog muaj sib npaug yog hais tias lawv coj sab thiab cov ces kaum muaj sib npaug. Nrog rau qhov no kab zauv, yog hais tias peb muab txim no ob nuj nqis ntawm txhua lwm yam, tag nrho lawv tej kab ntws los sib tshuam. Tsis tas li ntawd daim duab tej zaum yuav zoo sib xws, nyob rau hauv particular, nws txhawj xeeb tshuam zoo tib yam daim, differing xwb nyob rau hauv qhov ntau. Nyob rau hauv thiaj li yuav ua xws li ib tug xaus rau lub sawv cev cov vajvoos peb sab yuav tsum tau ntsib nyob rau hauv ib tug ntawm cov nram qab no tej yam kev mob:

• ob ces kaum ntawm ib daim duab yog sib npaug zos rau ob ces kaum ntawm lwm tus;
• xwm yeem mus rau ob tog ntawm lub ob sab ntawm lub thib ob daim duab peb sab, thiab lub ces kaum ntawm lub tsim sab yog muaj sib npaug;
• peb sab ntawm lub thib ob daim duab yog tib yam li hais tias ntawm lub ua ntej.

Ntawm cov hoob kawm, rau cov undisputed koob pheej ntawm lawv, uas tsis yog tus tsim slightest tsis ntseeg, koj yuav tsum muaj tib lub qhov tseem ceeb ntawm tag nrho cov ntsiab ntawm ob nuj nqis, tab sis, nrog rau qhov teeb meem ntawm txoj kev tshawb xav yog heev yooj yim zog, thiab tsuas yog ib ob peb yam kev mob tau tso cai rau muaj los ua pov thawj hais tias cov duab peb ceg.

Tus thawj kos npe rau ntawm koob pheej ntawm lawv ntawm cov vajvoos peb sab

rau lub npe teeb meem solved rau lub hauv paus ntawm cov ntawv pov thawj ntawm cov theorem, uas nyeem hais tias raws li nram no: "Yog hais tias ob sab ntawm daim duab peb sab thiab lub kaum sab xis uas lawv tsim, yog sib npaug zos rau ob sab thiab lub kaum sab xis ntawm lub lwm daim duab peb sab, ces cov nuj nqis yog tseem sib npaug zos rau txhua lwm yam."

Raws li lub suab pov thawj ntawm cov theorem txog thawj kos npe rau ntawm koob pheej ntawm lawv ntawm cov vajvoos peb sab? Sawv daws yeej paub hais tias ob theem yog muaj sib npaug yog hais tias lawv muaj tib lub ntev, los yog ib ncig sib npaug zos yog hais tias lawv muaj tib lub voos kheej-kheej. Thiab nyob rau hauv cov ntaub ntawv ntawm daim duab peb sab muaj ib tug ob peb cim uas nws yuav lam xav tias cov nuj nqis yog zoo tib yam, uas yog heev pab tau nyob rau hauv kev daws ntau yam duab teeb meem.

Lub suab ntawm lub theorem "Tus thawj kos npe rau ntawm koob pheej ntawm lawv ntawm cov vajvoos peb sab", piav saum toj no, tab sis nws pov thawj:

• Es daim duab peb sab ABC thiab A ₁ B ₁ C ₁ yog cov tib sab AB thiab A ₁ B ₁ thiab, ntsig txog, BC thiab B ₁ C _1, thiab lub ces kaum uas ua los ntawm cov sab muaj nqi, i.e. sib npaug zos. Ces muab tso rau nws nyob rau hauv ABC △ △ A ₁ B ₁ C _1, peb tau txais ib tug match ntawm tag nrho cov kab thiab vertices. Nws hais ntxiv hais tias cov duab peb ceg yog raws nraim tib yam, uas txhais tau tias sib npaug zos.

Theorem "Tus thawj kos npe rau ntawm koob pheej ntawm lawv ntawm cov vajvoos peb sab," kuj hu ua "Nyob rau ob sab thiab ces kaum." Ua tau, qhov no yog qhov essence ntawm nws.

Theorem rau lub thib ob kos npe rau

Qhov thib ob kos npe rau ntawm koob pheej ntawm lawv yog muaj pov thawj zoo, lub pov thawj raws li nyob rau hauv lub fact tias tus imposition ntawm daim rau txhua lwm yam, lawv yeej zoo tib yam nyob rau hauv tag nrho cov saum thiab sab. Ib tug theorem suab zoo li no: "Yog hais tias ib sab thiab ob ces kaum nyob rau hauv lub tsim ntawm cov uas nws koom, lub tog thiab tus ob ces kaum ntawm lub thib ob daim duab peb sab, ces cov nuj nqis yog zoo tib yam, piv txwv li sib npaug zos."

Qhov thib peb kos npe rau thiab pov thawj

Yog hais tias ob qho tag nrho lub 2 thiab lub 1 kos npe rau ntawm koob pheej ntawm lawv siv rau ob tog ntawm lub duab peb ceg, ces kaum thiab nrhiav, peb hais txog rau cov ob tog. Yog li, lub theorem muaj cov nram qab no hais: "Yog hais tias tag nrho cov sab ntawm ib lub duab peb sab yog muaj sib npaug mus rau peb sab ntawm lub thib ob daim duab peb sab, cov nuj nqis yog zoo tib yam."

Los ua pov thawj no theorem, nws yog tsim nyog los delve rau hauv ntau dua kom meej nyob rau hauv lub ntsiab txhais ntawm koob pheej ntawm lawv. Nyob rau hauv qhov tseeb, yog dab tsi yog meant by "cov vajvoos peb sab yog muaj sib npaug"? Yog leejtwg hais tias yog hais tias peb muab txim ib daim duab mus rau lwm lub, tag nrho cov ntsiab phim, nws tsuas yuav cov ntaub ntawv thaum lawv sab thiab cov ces kaum muaj sib npaug. Nyob rau tib lub sij hawm lub kaum sab xis opposite rau ib sab, uas yog tib yam li rau lwm yam daim duab peb sab yog sib npaug zos rau cov coj kab los sib ntsib ntawm lub thib ob daim duab. Nws yuav tsum tau muab sau tseg tias nyob rau hauv no taw tes rau hauv lub pov thawj yog ib qho yooj yim mus txhais ua 1 kos npe rau ntawm koob pheej ntawm lawv ntawm cov vajvoos peb sab. Yog hais tias qhov no ib theem zuj zus yog tsis pom, lub koob pheej ntawm lawv ntawm cov vajvoos peb sab tsuas yog tsis yooj yim sua, tsuas yog nyob rau hauv tus neeg mob qhov twg hauv daim duab yog ib daim iav duab ntawm cov thawj.

Txoj cai cov vajvoos peb sab

Tus qauv ntawm xws cov vajvoos peb sab yog ib txwm tus kab los sib ntsib nrog lub kaum sab xis 90 °. Yog li ntawd, cov lus nram qab no muaj tseeb:

• cov vajvoos peb sab nrog rau txoj kev lub kaum sab xis yog muaj sib npaug yog hais tias tus ceg ntawm lub thib ob cathetus zoo tib yam;
• cov nuj nqis yog muaj sib npaug yog hais tias lawv yog muaj sib npaug mus rau lub hypotenuse thiab ib tug ntawm ob txhais ceg;
• xws cov vajvoos peb sab yog muaj sib npaug yog hais tias lawv ob txhais ceg thiab zoo tib yam mob kaum sab xis.

Qhov no feature koom lub duab peb ceg. Los ua pov thawj Theorem siv app nrhiav kom txhua lwm yam, uas ua nyob rau hauv ob txhais ceg ntawm lub duab peb ceg yog daim ntau quav tso li hais tias ob tug ncaj sab laug ncaj lub nrog CA ₁ thiab CA sab.

ua daim ntawv thov

Nyob rau hauv Feem ntau, nyob rau hauv kev xyaum, nws thov cov thawj kos npe rau ntawm koob pheej ntawm lawv ntawm cov vajvoos peb sab. Nyob rau hauv qhov tseeb, qhov no seemingly yooj yim hauv chav kawm ntawv rau geometry thiab dav hlau geometry siv lub ntsiab thiab 7 mus rau xam qhov ntev, piv txwv li, tus xov tooj cable tsis muaj ib tug ntsuas cheeb tsam, nyob rau hauv uas nws yuav muab qhov chaw. Siv no theorem nws yog ib qho yooj yim los mus ua kom cov kev tsim nyog suav los mus txiav txim qhov ntev ntawm cov kob, nyob rau hauv nruab nrab ntawm tus dej, tsis muaj dej nyob nws. Los yog txhawb lub laj kab los ntawm ib tse nrog koj bar nyob rau hauv lub bay li hais tias nws yog muab faib ua ob sib npaug zos cov vajvoos peb sab, los yog xam cov complex cov ntsiab ntawm cov ua hauj lwm nyob rau hauv ntoo los yog nyob rau hauv lub xam ntawm truss ru tsev system thaum lub sij hawm tsim kho.

Tus thawj kos npe rau ntawm koob pheej ntawm lawv ntawm cov vajvoos peb sab muaj ntau daim ntawv thov nyob rau hauv ib tug tiag tiag "neeg laus" lub neej. Thaum nyob rau hauv high school xyoo nws yog ib lub npe rau ntau mas boring thiab lig ruaj.