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Tứ giác lồi

Tứ giác là gì? Tứ giác lồi là gì?
Quảng cáo

1. Lý thuyết

Khái niệm:

+ Tứ giác 🔯là hình gồm bốn đoạn thẳng, trong đó không có hai đoạn thẳng nào cùng nằm trên một đường thẳng.

+ Tứ giác lồi 🐷là tứ giác mà hai đỉnh thuộc một cạnh bất kì luôn nằm về một phía của đường thẳng đi qua hai đỉnh còn lại.

2. Ví dụ minh họa

- Ví dụ: Trong các hình dưới đây, hình 1 được gọi là tứ giác lồi.

 

- Chú ý: Dạng bài toán thường gặp: Bài toán liên quan đến cạnh và đường chéo của tứ giác lồi

Phương pháp giải:

Ta có thể chia tứ giác thành các tam giác, sau đó vận dụng bất đẳng thức tam giác.

Ví dụ: 🔯Cho tam giác ABCD. Chứng minh \(AC + BD > \frac{p}{2}\) (p là chu vi tứ giác).

Lời giải:

Gọi I là giao điểm của AC và BD. Theo bất đẳng thức tam giác ta có:\(IA + IB > AB\);\(IB + IC > BC\);\(IC + ID > CD\);\(ID + IA > AD\).Cộng từng vế, ta được:\(IA + IB + IB + IC + IC + ID + ID + IA > AB + BC + CD + DA\)\(2(IA + IB + IC + ID) > AB + BC + CD + DA\)\(2[(IA + IB) + (IC + ID)] > p\)\(2(AC + BD) > p\)\(AC + BD > \frac{p}{2}\) (đpcm)

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