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Định lí Thales trong tam giác

Định lí Thales là gì? Định lí Thales đảo là gì? Hệ quả của định lí Thales là gì?
Quảng cáo

1. Lý thuyết

- Định lí Thalès:﷽ Nếu một đường thẳng song song với một cạnh của tam giác và cắt hai cạnh còn lại thì nó định ra trên hai cạnh đó những đoạn thẳng tương ứng tỉ lệ.

GT \(\Delta ABC,B'C'//BC(B' \in AB,C' \in AC)\)
KL \(\frac{{AB'}}{{AB}} = \frac{{AC'}}{{AC}};\frac{{AB'}}{{B'B}} = \frac{{AC'}}{{C'C}};\frac{{B'B}}{{AB}} = \frac{{C'C}}{{AC}}\)

- Định lí Thalès đảo: ꦡNếu một đường thẳng cắt hai cạnh của một tam giác và định ra trên hai cạnh này những đoạn thẳng tương ứng tỉ lệ thì đường thẳng đó song song với cạnh còn lại của tam giác.

GT \(\Delta ABC,D \in AB,E \in AC,\frac{{AD}}{{AB}} = \frac{{AE}}{{AC}}\)  hoặc \(\frac{{AD}}{{BD}} = \frac{{AE}}{{CE}}\) hoặc \(\frac{{BD}}{{AB}} = \frac{{CE}}{{AC}}\)
KL \(DE//BC\)

- Hệ quả của định lí Thalès:✱ Nếu một đường thẳng cắt hai cạnh của một tam giác và song song với cạnh thứ ba thì tạo ra một tam giác mới có ba cạnh tương ứng tỉ lệ với ba cạnh của tam giác đã cho.

GT \(\Delta ABC,B'C'//BC(B' \in AB,C' \in AC)\)
KL \(\frac{{AB'}}{{AB}} = \frac{{AC'}}{{AC}} = \frac{{B'C'}}{{BC}}\)

Chú ý: 🔯Hệ quả trên vẫn đúng cho trường hợp đường thẳng a song song với một cạnh của tam giác và cắt phần kéo dài của hai cạnh còn lại.

 

Ở hai hình trên, tam giác ABC có BC // B’C’ \( \Rightarrow \frac{{AB'}}{{AB}} = \frac{{AC'}}{{AC}} = \frac{{B'C'}}{{BC}}\).

2. Ví dụ minh họa

- Ví dụ về Định lí Thales:

 

Tam giác ABC, DE // BC \( \Rightarrow \frac{{AD}}{{AB}} = \frac{{AE}}{{AC}}\) và \(\frac{{AD}}{{DB}} = \frac{{AE}}{{EC}}\).

- Ví dụ về Định lí Thales đảo:

 

Tam giác ABC có \(\frac{{AD}}{{DB}} = \frac{{AE}}{{EC}} \Rightarrow DE//BC\).

- Ví dụ về Hệ quả của định lí Thales:

 

Tam giác ABC, DE // BC \( \Rightarrow \frac{{AD}}{{AB}} = \frac{{AE}}{{AC}} = \frac{{DE}}{{BC}}\).

Quảng cáo

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