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Mai Thị Thúy
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Đặng Thị Hạnh
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Phạm Thảo Vân
9 tháng 4 2016 lúc 16:19

B A K H C E I D

Ta có \(\widehat{AHC}=\widehat{AEC}=90^0\) nên 4 điểm A, H, C, E cùng thuộc đường tròn đường kính AC.

Gọi I là giao điểm của AC và BD

Ta có \(\widehat{HIE}=2\widehat{HAE}=2\left(180^0-\widehat{BCD}\right)\)

Các tứ giác AKED, AKHB nội tiếp nên \(\widehat{EKD}=\widehat{EAD}\) và \(\widehat{BKH}=\widehat{BAH}\)

Do đó \(\widehat{HKE}=180^0-\widehat{AKD}-\overrightarrow{BKH}=180^0-\overrightarrow{EAD}-\overrightarrow{BAH}=2\overrightarrow{HAE}=2\left(180^0-\overrightarrow{BCD}\right)=\overrightarrow{HIE}\)

Vậy tứ giác HKIE nội tiếp. Do đó I thuộc đường tròn (C) ngoại tiếp tam giác HKE

- Gọi \(C\left(c;c-3\right)\in d\left(c>0\right)\Rightarrow I\left(\frac{c-2}{2};\frac{c-4}{2}\right)\)

Do I thuộc (C) nên có phương trình :

\(c^2-c-2=0\Leftrightarrow c=2\) V c=-1 (loại c=-1) Suy ra \(C\left(2;-1\right);I\left(0;-1\right)\)

- Điểm E, H nằm trên đường tròn đường kính AC và đường tròn (C) nên tọa độ thỏa mãn hệ phương trình :

\(\begin{cases}x^2+y^2+x+4y+3=0\\x^2+\left(y+1\right)^2=4\end{cases}\) \(\Leftrightarrow\begin{cases}x=0;y=-3\\x=-\frac{8}{5};y=-\frac{11}{2}\end{cases}\)

- Vì H có hoành độ âm nên \(H\left(-\frac{8}{5};-\frac{11}{5}\right);E\left(0;-3\right)\) Suy ra \(AB:x-y+1=0;BC:x-3y-5=0\)

Tọa độ B thỏa mãn \(\begin{cases}x-y+1=0\\x-3y-5=0\end{cases}\) \(\Leftrightarrow B\left(-4;-3\right)\Rightarrow\overrightarrow{BA}=\left(2;2\right);\overrightarrow{BC}=\left(6;2\right)\Rightarrow\overrightarrow{BA}.\overrightarrow{BC}=16>0\)

Vì \(\overrightarrow{AB}=\overrightarrow{DC}\Rightarrow D\left(4;1\right)\)

Vậy \(B\left(-4;-3\right);C\left(2;-1\right);D\left(4;1\right)\)

dsfdsf
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Trịnh Việt Dũng
15 tháng 6 2022 lúc 20:31

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Pham Trong Bach
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Cao Minh Tâm
2 tháng 9 2019 lúc 5:46

Cindy
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Lê Mạnh Cường
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vo nhi
25 tháng 4 2018 lúc 20:00

de ***** tu lam dihihi

Trần Hà Ngân Khánh
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Thảo Vi
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Nguyễn Việt Lâm
29 tháng 1 2021 lúc 15:40

1.

\(\overrightarrow{AB}=\left(2;-6\right)\Rightarrow AB=2\sqrt{10}\) \(\Rightarrow BC=AB.cosB=\sqrt{10}\)

Gọi \(C\left(x;y\right)\Rightarrow\left\{{}\begin{matrix}\overrightarrow{AC}=\left(x-1;y-2\right)\\\overrightarrow{BC}=\left(x-3;y+4\right)\end{matrix}\right.\)

Tam giác ABC vuông tại C và có \(BC=\sqrt{10}\)

\(\Leftrightarrow\left\{{}\begin{matrix}\overrightarrow{AC}.\overrightarrow{BC}=0\\BC^2=10\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}\left(x-1\right)\left(x-3\right)+\left(y-2\right)\left(y+4\right)=0\\\left(x-3\right)^2+\left(y+4\right)^2=10\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x^2+y^2-4x+2y-5=0\\x^2+y^2-6x+8y+15=0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x-3y-10=0\\x^2+y^2-6x+8y+15=0\end{matrix}\right.\)

\(\Rightarrow\left(3y+10\right)^2+y^2-6\left(3y+10\right)+8y+15=0\)

\(\Leftrightarrow2y^2+10y+11=0\)

\(\Leftrightarrow y=...\)

Nguyễn Việt Lâm
29 tháng 1 2021 lúc 16:00

2.

Kẻ \(EF\perp BC\)

\(S_{ABC}=9S_{BDE}\Rightarrow AD.BC=9EF.BD\Rightarrow\dfrac{EF}{AD}=\dfrac{BC}{9BD}\)

Talet: \(\dfrac{EF}{AD}=\dfrac{BF}{BD}=\dfrac{BC}{9BD}\Rightarrow BC=9BF\)

Hệ thức lượng: \(BE^2=BF.BC=9BF^2\Rightarrow BE=3BF\)

\(\Rightarrow cosB=\dfrac{BF}{BE}=\dfrac{1}{3}\)

Gọi R là bán kính đường tròn ngoại tiếp ABC và \(r\) là bán kính đường tròn ngoại tiếp BDE

\(sinB=\sqrt{1-\left(\dfrac{1}{3}\right)^2}=\dfrac{2\sqrt{2}}{3}\)

\(\Rightarrow r=\dfrac{DE}{2sinB}=\dfrac{3}{2}\) (định lý sin tam giác BDE)

Dễ dàng chứng minh 2 tam giác ABC và BDE đồng dạng (chung góc B và \(\widehat{A}=\widehat{BDE}\) vì cùng bù \(\widehat{CDE}\))

Mà \(S_{ABC}=9S_{BDE}\Rightarrow\) 2 tam giác đồng dạng tỉ số \(k=\sqrt{9}=3\)

\(\Rightarrow R=3r=\dfrac{9}{2}\)

Nguyễn Việt Lâm
29 tháng 1 2021 lúc 16:01

Hình vẽ bài số 2:

undefined

Pham Trong Bach
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Cao Minh Tâm
11 tháng 10 2019 lúc 8:01

Ta có: AB → = (−a; b; 0) và  AC →  = (−a; 0; c)

Vì  AB → .  AC →  = a 2 > 0 nên góc BAC là góc nhọn.

Lập luận tương tự ta chứng minh được các góc  ∠ B và  ∠ C cũng là góc nhọn.