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Lã Nguyễn Gia Hy
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Lã Nguyễn Gia Hy
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dsfdsf
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Phạm Thị Thùy Hương
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%Hz@
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Kiệt Nguyễn
13 tháng 4 2020 lúc 20:57

Gọi H, K lần lượt là trung điểm của BC, C'A'

\(\Delta A'BC'\)cân tại B có \(\widehat{A'BC'}=120^0\)\(\Rightarrow\widehat{BC'A'}=\widehat{BA'C'}=30^0\)

\(\Rightarrow\Delta BKC'\)là nửa tam giác đều

\(\Rightarrow BK=\frac{1}{2}BC'\)(1)

\(AH\perp BC\)(do \(\Delta ABC\)đều) nên \(\Delta ABH\)là nửa tam giác đều

\(\Rightarrow BH=\frac{1}{2}AB\)(2)

Từ (1) và (2) suy ra \(\frac{BK}{BC'}=\frac{BH}{AB}\)

Ta có: \(\widehat{KBH}=60^0-\widehat{ABK}=\widehat{ABC'}\)

 \(\Delta KBH\)và \(\Delta C'BA\)có: \(\frac{BK}{BC'}=\frac{BH}{BA}\left(cmt\right)\)\(\widehat{KBH}=\widehat{C'BA}\left(cmt\right)\)

 \(\Rightarrow\Delta KBH~\Delta C'BA\left(c-g-c\right)\)

\(\Rightarrow\frac{KH}{C'A}=\frac{1}{2}\Rightarrow\frac{KH}{AB'}=\frac{1}{2}\)và \(\widehat{C'AB}=\widehat{KHB}\)

Ta có: \(\widehat{HAB'}=\widehat{B'AC'}-\left(30^0+\widehat{C'AB}\right)\)

\(=\left(\widehat{B'AC'}-30^0\right)-\widehat{C'AB}=90^0-\widehat{KHB}=\widehat{KHA}\)

Mà \(\widehat{HAB'}\)và \(\widehat{KHA}\)ở vị trí so le trong nên KH // AB'

\(\Rightarrow\frac{KG}{GB'}=\frac{GH}{GA}=\frac{KH}{AB'}=\frac{1}{2}\)

hay \(\frac{B'G}{KB'}=\frac{GA}{HA}=\frac{2}{3}\)

Điều này chứng tỏ \(\Delta ABC\)và \(\Delta A'B'C'\)có cùng trọng tâm (đpcm)

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Vũ Như Ý
13 tháng 4 2020 lúc 22:01

dễ quá

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Hà Thúy Nga
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Nguyen Duc Thong
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Nguyễn S
23 tháng 1 2022 lúc 17:44

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