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Gọi M là trung điểm BC ; N là điểm đối xứng với H qua M.
M là trung điểm của BC và HN nên BNCH là hình bình hành
\(\Rightarrow NC//BH\)
Mà \(BH\perp AC\Rightarrow NC\perp AC\)hay AN là đường kính của đường tròn ( O )
Dễ thấy OM là đường trung bình \(\Delta AHN\) suy ra \(OM=\frac{1}{2}AH\)
M là trung điểm BC nên OM \(\perp\)BC
Xét \(\Delta AHG\)và \(\Delta OGM\)có :
\(\widehat{HAG}=\widehat{GMO}\); \(\frac{GM}{GA}=\frac{OM}{HA}=\frac{1}{2}\)
\(\Rightarrow\Delta AGH~\Delta MOG\left(c.g.c\right)\Rightarrow\widehat{AGH}=\widehat{MGO}\)hay H,G,O thẳng hàng
gọi E,F,T lần lượt là trung điểm của AB,CD,BD
Đường thẳng ME cắt NF tại S
Vì AC = BD \(\Rightarrow EQFP\)là hình thoi \(\Rightarrow EF\perp PQ\)( 1 )
Xét \(\Delta TPQ\)và \(\Delta SEF\)có : \(ME\perp AB,TP//AB\)
Tương tự , \(NF\perp CD;\)\(TQ//CD\)
\(\Rightarrow\Delta TPQ~\Delta SEF\)( Góc có cạnh tương ứng vuông góc )
\(\Rightarrow\frac{SE}{SF}=\frac{TP}{TQ}=\frac{AB}{CD}\)
Mặt khác : \(\Delta MAB~\Delta NCD\Rightarrow\frac{AB}{CD}=\frac{ME}{NF}\)( tỉ số đường cao = tỉ số đồng dạng )
Suy ra : \(\frac{ME}{NF}=\frac{SE}{SF}\)\(\Rightarrow EF//MN\)( 2 )
Từ ( 1 ) và ( 2 ) suy ra \(MN\perp PQ\)
Gọi giao điểm của hai đường chéo là O giao điểm của hai cạnh bên là S,giao điểm của SO với AB,CD lần lượt là X,Y.
Ta có AX//YC nên theo định lý Ta lét ta có:
\(\frac{AX}{YC}\)=\(\frac{AO}{OC}\)=\(\frac{AB}{DC}\)=\(\frac{AX}{DY}\)
=>YC=DY
Vậy Y là trung điểm của DC.
Ta có AB//DC theo định lý Ta-lét ta có:
\(\frac{AX}{DY}\)=\(\frac{SX}{XY}\)=\(\frac{XB}{YC}\)
mà DY=YC(c/m trên)
=>AX=XB=>X là trung điểm của AB
Vậy giao điểm của SO với AB,CD tại trung điểm của các cạnh đó
=>đpcm
Ta cũng dễ dàng chứng mình được đường thẳng chứa 4 điểm đó là trùng trực của hai cạnh đấy sao khi chừng minh chúng thẳng hàng ở trên nhé!
Gọi giao điểm của hai đường chéo là O giao điểm của hai cạnh bên là S,giao điểm của SO với AB,CD lần lượt là X,Y.
Ta có AX//YC nên theo định lý Ta lét ta có:
AXYCAXYC=AOOCAOOC=ABDCABDC=AXDYAXDY
=>YC=DY
Vậy Y là trung điểm của DC.
Ta có AB//DC theo định lý Ta-lét ta có:
AXDYAXDY=SXXYSXXY=XBYCXBYC
mà DY=YC(c/m trên)
=>AX=XB=>X là trung điểm của AB
Vậy giao điểm của SO với AB,CD tại trung điểm của các cạnh đó
=>đpcm
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Bài 1:
a: Ta có: ΔBKC vuông tại K
mà KM là đường trung tuyến
nên KM=BC/2(1)
Ta có: ΔBHC vuông tại H
mà HM là đường trung tuyến
nên HM=BC/2(2)
Từ (1)và (2) suy ra MH=MK
hay ΔMHK cân tại M
b: Kẻ MN vuông góc với HK
=>N là trung điểm của HK
Xét hình thang CBDE có
M là trung điểm của BC
MN//DB//EC
DO đó: N là trung điểm của DE
=>DK=HE