Question

[Nkokmt]

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.