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a) Ta có: \(\left\{{}\begin{matrix}\widehat{BAC}+\widehat{EAB}=\widehat{EAC}\\\widehat{BAC}+\widehat{HAC}=\widehat{BAH}\end{matrix}\right.\) (1)
ABDE là hình vuông
=> Góc EAB = 900 (2)
Và: AE = AB
ACGH là hình vuông
=> Góc HAC = 900 (3)
Và: AH = AC
Từ (1); (2) và (3)
=> Góc EAC = Góc BAH
Xét ΔAEC và ΔABH ta có
AC = AH (cmt)
Góc EAC = Góc BAH (cmt)
AE = AB (cmt)
=> ΔAEC = ΔABH (c - g - c)
=> EC = BH (2 cạnh tương ứng) (4)
b) Có I là tâm của hình vuông ABDE
=> I là trung điểm của đường chéo EB
Lại có: M là trung điểm của BC
=> IM là đường TB của tam giác EBC
\(\Rightarrow IM=\frac{1}{2}EC\left(5\right)\)
Có J là tâm của hình vuông ACGH
=> J là trung điểm của đường chéo HC
Lại cos: M là trung điểm của BC
=> JM là đường TB của tam giác HBC
\(\Rightarrow JM=\frac{1}{2}BH\left(6\right)\)
Từ (4); (5); (6)
=> JM = IM
=> Tam giác JIM cân tại M (*)
Tự chứng minh EC ⊥ BH
Lại có:
+) JM là đường TB của tam giác HBC (cmt) => JM // BH
+) IM là đường TB của tam giác EBC (cmt) => IM // EC
=>IM vuông góc với JM (**)
Từ (*) và (**) => đpcm
Đề bị sai từ câu đầu tiên. Tam giác chỉ có thể kí hiệu bằng 3 chữ cái in hoa.
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