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Cho điểm A chạy trên nửa đường tròn đường kính BC cố định.

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Đề bài

Cho điểm A chạy trên nửa đường tròn đường kính BC cố định. Vẽ về phía ngoài tam giác ABC hình vuông ABEF. Chứng minh rằng điểm E chạy trên một nửa đường tròn cố định.

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Lời giải chi tiết

Gọi O là tâm của đường tròn đường kính BC.Vì ABEF là hình vuông nên \(BA{\rm{ }} = {\rm{ }}BE\)  và \(\left( {BA,BE} \right) = \widehat {ABE} = {90^o}\)Suy ra phép quay tâm B, góc quay 90° biến điểm A thành điểm E.Đặt \(C'{\rm{ }} = {\rm{ }}{Q_{\left( {B,{\rm{ }}90^\circ } \right)}}\left( C \right)\) và \(O'{\rm{ }} = {\rm{ }}{Q_{\left( {B,{\rm{ }}90^\circ } \right)}}\left( O \right).\)Ta có \(B{\rm{ }} = {\rm{ }}{Q_{\left( {B,{\rm{ }}90^\circ } \right)}}\left( B \right).\)Vậy khi điểm A chạy trên nửa đường tròn tâm O, đường kính BC cố định thì điểm E chạy trên nửa đường tròn tâm O’, đường kính BC’ cố định là ảnh của nửa đường tròn tâm O, đường kính BC qua phép quay tâm B, góc quay 90°.

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