Cho tam giác $ABC$ có B^=C^" role="presentation" style="border:0px; direction:ltr; display:inline-block; float:none; font-family:arial,tahoma,helvetica,sans-serif; font-size:18.72px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:1px 0px; position:relative; white-space:nowrap" class="MathJax_CHTML mjx-chtml">ˆB=ˆC$\hat{B}=\hat{C}$. Tia phân giác góc $A$ cắt $BC$ tại $D.$ Chứng minh rằng $DB=DC,AB=AC.$

Cho tam giác $ABC.$ Các tia phân giác của các góc $B$ và $C$ cắt nhau ở $O.$ Kẻ OD⊥AC" role="presentation" style="border:0px; direction:ltr; display:inline-block; float:none; font-family:arial,tahoma,helvetica,sans-serif; font-size:18.72px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:1px 0px; position:relative; white-space:nowrap" class="MathJax_CHTML mjx-chtml">OD⊥AC$\mathrm{O}\mathrm{D}\mathrm{\perp }AC$, kẻ OE⊥AB" role="presentation" style="border:0px; direction:ltr; display:inline-block; float:none; font-family:arial,tahoma,helvetica,sans-serif; font-size:18.72px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:1px 0px; position:relative; white-space:nowrap" class="MathJax_CHTML mjx-chtml">OE⊥AB$\mathrm{O}E\mathrm{\perp }AB$. Chứng minh rằng $OD=OE.$

Cho tam giác $ABC$ có B^=C^=50∘" role="presentation" style="border:0px; direction:ltr; display:inline-block; float:none; font-family:arial,tahoma,helvetica,sans-serif; font-size:18.72px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:1px 0px; position:relative; white-space:nowrap" class="MathJax_CHTML mjx-chtml">ˆA=ˆB=60∘$\hat{B}=\hat{C}={50}^{\circ }$. Gọi tia $Am$ là tia phân giác của góc ngoài ở đỉnh $\mathrm{C.}$ Hãy chứng tỏ //BC" role="presentation" style="border:0px; direction:ltr; display:inline-block; float:none; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:1px 0px; position:relative; white-space:nowrap" class="MathJax_CHTML mjx-chtml">Cx/
/AB.

Bài 1. Cho tam giác ABC, điểm D thuộc cạnh BC. Gọi M là trung điểm của AD. Trên tia đối của tia MB lấy điểm E sao cho ME = MB. Trên tia đối của tia MC lấy F sao cho MF = MC. Chứng minh:

a) AE = BD;

b) AF // BC.

c) Ba điểm A, E, F thẳng hàng.

Bài 2. Cho tam giác ABC cân tại A, M là trung điểm của BC.

a) Chứng minh AFE&#x005E;=ABC&#x005E;&#x21D2;EF//BC" role="presentation" style="border:0px; direction:ltr; display:inline-block; float:none; font-size:18.08px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:1px 0px; position:relative; text-align:left; white-space:nowrap; word-spacing:normal" class="MathJax_CHTML mjx-chtml">ˆAFE=ˆABC⇒EF//BCAFE&#x005E;=ABC&#x005E;&#x21D2;EF//BC" role="presentation" style="border:0px; direction:ltr; display:inline-block; float:none; font-size:18.08px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:1px 0px; position:relative; text-align:left; white-space:nowrap; word-spacing:normal" class="MathJax_CHTML mjx-chtml">$\stackrel{<u><em>^</em></u>}{\mathrm{<u><em>A</em></u>}\mathrm{<u><em>F</em></u>}\mathrm{<u><em>E</em></u>}}<u><em>=</em></u>\stackrel{<u><em>^</em></u>}{\mathrm{<u><em>A</em></u>}\mathrm{<u><em>B</em></u>}\mathrm{<u><em>C</em></u>}}<u><em>\Rightarrow </em></u>\mathrm{<u><em>E</em></u>}\mathrm{<u><em>F</em></u>}<u><em>/</em></u><u><em>/</em></u>\mathrm{<u><em>B</em></u>}\mathrm{<u><em>C</em></u>}$ &#x0394;ABM=&#x0394;ACM" role="presentation" style="border:0px; direction:ltr; display:inline-block; float:none; font-size:18.08px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:1px 0px; position:relative; text-align:left; white-space:nowrap; word-spacing:normal" class="MathJax_CHTML mjx-chtml">ΔABM=ΔACM&#x0394;ABM=&#x0394;ACM" role="presentation" style="border:0px; direction:ltr; display:inline-block; float:none; font-size:18.08px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:1px 0px; position:relative; text-align:left; white-space:nowrap; word-spacing:normal" class="MathJax_CHTML mjx-chtml">$\mathrm{<u><em>\Delta </em></u>}\mathrm{<u><em>A</em></u>}\mathrm{<u><em>B</em></u>}\mathrm{<u><em>M</em></u>}<u><em>=</em></u>\mathrm{<u><em>\Delta </em></u>}\mathrm{<u><em>A</em></u>}\mathrm{<u><em>C</em></u>}\mathrm{<u><em>M</em></u>}$.

b) Chứng minh &#x22A5;BC." role="presentation" style="border:0px; direction:ltr; display:inline-block; float:none; font-size:18.08px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:1px 0px; position:relative; text-align:left; white-space:nowrap; word-spacing:normal" class="MathJax_CHTML mjx-chtml">AM⊥BC.&#x22A5;BC." role="presentation" style="border:0px; direction:ltr; display:inline-block; float:none; font-size:18.08px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:1px 0px; position:relative; text-align:left; white-space:nowrap; word-spacing:normal" class="MathJax_CHTML mjx-chtml">$\mathrm{<u><em>A</em></u>}\mathrm{<u><em>M</em></u>}\mathrm{<u><em>\perp </em></u>}\mathrm{<u><em>B</em></u>}\mathrm{<u><em>C</em></u>}<u><em>.</em></u>$

c) Trên cạnh BA lấy  điểm E. Trên cạnh CA lấy điểm F sao cho BE = CF. Chứng minh &#x0394;EBC" role="presentation" style="border:0px; direction:ltr; display:inline-block; float:none; font-size:18.08px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:1px 0px; position:relative; text-align:left; white-space:nowrap; word-spacing:normal" class="MathJax_CHTML mjx-chtml">ΔEBC&#x0394;EBC" role="presentation" style="border:0px; direction:ltr; display:inline-block; float:none; font-size:18.08px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:1px 0px; position:relative; text-align:left; white-space:nowrap; word-spacing:normal" class="MathJax_CHTML mjx-chtml">$\mathrm{<u><em>\Delta </em></u>}\mathrm{<u><em>E</em></u>}\mathrm{<u><em>B</em></u>}\mathrm{<u><em>C</em></u>}$ và &#x0394;FCB" role="presentation" style="border:0px; direction:ltr; display:inline-block; float:none; font-size:18.08px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:1px 0px; position:relative; text-align:left; white-space:nowrap; word-spacing:normal" class="MathJax_CHTML mjx-chtml">ΔFCB&#x0394;FCB" role="presentation" style="border:0px; direction:ltr; display:inline-block; float:none; font-size:18.08px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:1px 0px; position:relative; text-align:left; white-space:nowrap; word-spacing:normal" class="MathJax_CHTML mjx-chtml">$\mathrm{<u><em>\Delta </em></u>}\mathrm{<u><em>F</em></u>}\mathrm{<u><em>C</em></u>}\mathrm{<u><em>B</em></u>}$bằng nhau.

d) Chứng minh EF // BC.

Cho biết hai đại lượng $x$ và $y$ tỉ lệ nghịch với nhau. Điền số thích hợp vào ô trống sau đây:

$Cho$ &#x0394;MNP" role="presentation" style="border:0px; box-sizing:border-box; direction:ltr; display:inline-block; float:none; font-size:20.34px; font-style:normal; font-weight:normal; letter-spacing:normal; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:1px 0px; position:relative; text-align:left; text-indent:0px; text-transform:none; white-space:nowrap; word-spacing:normal" class="MathJax_CHTML mjx-chtml">ΔMNP$\mathrm{\Delta }MNP$ &#xF3;" role="presentation" style="border:0px; box-sizing:border-box; direction:ltr; display:inline-block; float:none; font-size:20.34px; font-style:normal; font-weight:normal; letter-spacing:normal; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:1px 0px; position:relative; text-align:left; text-indent:0px; text-transform:none; white-space:nowrap; word-spacing:normal" class="MathJax_CHTML mjx-chtml">có$có$ M&#x005E;=30o" role="presentation" style="border:0px; box-sizing:border-box; direction:ltr; display:inline-block; float:none; font-size:20.34px; font-style:normal; font-weight:normal; letter-spacing:normal; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:1px 0px; position:relative; text-align:left; text-indent:0px; text-transform:none; white-space:nowrap; word-spacing:normal" class="MathJax_CHTML mjx-chtml">ˆM=30o$\hat{M}={30}^o$$;$ N&#x005E;=20o" role="presentation" style="border:0px; box-sizing:border-box; direction:ltr; display:inline-block; float:none; font-size:20.34px; font-style:normal; font-weight:normal; letter-spacing:normal; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:1px 0px; position:relative; text-align:left; text-indent:0px; text-transform:none; white-space:nowrap; word-spacing:normal" class="MathJax_CHTML mjx-chtml">ˆN=20o$\hat{N}={20}^o$$.$ &#xEA;n" role="presentation" style="border:0px; box-sizing:border-box; direction:ltr; display:inline-block; float:none; font-size:20.34px; font-style:normal; font-weight:normal; letter-spacing:normal; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:1px 0px; position:relative; text-align:left; text-indent:0px; text-transform:none; white-space:nowrap; word-spacing:normal" class="MathJax_CHTML mjx-chtml">Trên$Trên$ $MN$ &#x1EA5;y" role="presentation" style="border:0px; box-sizing:border-box; direction:ltr; display:inline-block; float:none; font-size:20.34px; font-style:normal; font-weight:normal; letter-spacing:normal; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:1px 0px; position:relative; text-align:left; text-indent:0px; text-transform:none; white-space:nowrap; word-spacing:normal" class="MathJax_CHTML mjx-chtml">lấy$lấy$ $D$ $sao$ $cho$ $MD=NP$$.$ &#xED;nh" role="presentation" style="border:0px; box-sizing:border-box; direction:ltr; display:inline-block; float:none; font-size:20.34px; font-style:normal; font-weight:normal; letter-spacing:normal; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:1px 0px; position:relative; text-align:left; text-indent:0px; text-transform:none; white-space:nowrap; word-spacing:normal" class="MathJax_CHTML mjx-chtml">Tính$Tính$ NPD&#x005E;" role="presentation" style="border:0px; box-sizing:border-box; direction:ltr; display:inline-block; float:none; font-size:20.34px; font-style:normal; font-weight:normal; letter-spacing:normal; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:1px 0px; position:relative; text-align:left; text-indent:0px; text-transform:none; white-space:nowrap; word-spacing:normal" class="MathJax_CHTML mjx-chtml">ˆNPD$\widehat{NPD}$ $.$

Cho a,b&#x2208;R;n&#x2208;N&#x2217;" role="presentation" style="border:0px; color:rgb(40, 40, 40); direction:ltr; display:inline-block; float:none; font-family:helvetica,arial,sans-serif; font-size:16.38px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; padding:1px 0px; position:relative; white-space:nowrap; word-wrap:normal" class="MathJax_CHTML mjx-chtml">a,b∈R;n∈N∗${\displaystyle a,b\in R;n\in N^{\ast }}$. Chứng minh rằng:

 Cho biết $3$ người làm cỏ một cánh đồng hết $6$ giờ. Hỏi $12$ người (với cùng năng suất như thế) làm cỏ cánh đồng đó hết bao nhiêu thời gian?

Cho hình 56, trong đó //HK,AH//BK." role="presentation" style="border:0px; direction:ltr; display:inline-block; float:none; font-size:18.72px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:1px 0px; position:relative; white-space:nowrap; word-spacing:normal" class="MathJax_CHTML mjx-chtml">AB//HK,AH//BK.$AB//HK,AH//BK.$ Chứng minh rằng $AB=HK,AH=BK.$