Dựng đường trung trực của một đoạn thẳng
Geometry construction using a compass and straightedge
Bài toán dựng hình này chỉ ra cách dựng đường trung trực của một đoạn thẳng đã cho bằng thước kẻ và compass. Đường này có hai tính chất: (i) Cắt đoạn thẳng đã cho tại trung điểm của nó và (ii) vuông góc với đoạn thẳng tại trung điểm của nó. Cách dựng được mô tả như sau:
Bản in các bước dựng hình
The above animation is available as a printable step-by-step instruction sheet, which can be used for making handouts or when a computer is not available.
Chứng minh
This construction works by effectively building congruent triangles that result in right angles being formed at the midpoint of the line segment. The proof is surprisingly long for such a simple construction.
The image below is the final drawing above with the red lines and dots added to some angles.
| Argument | Reason | |
|---|---|---|
| 1 | Line segments AP, AQ, PB, QB are all congruent | The four distances were all drawn with the same compass width c. |
| Next we prove that the top and bottom triangles are isosceles and congruent | ||
| 2 | Triangles ∆APQ and ∆BPQ are isosceles | Two sides are congruent (length c) |
| 3 | Angles AQJ, APJ are congruent | Base angles of isosceles triangles are congruent |
| 4 | Triangles ∆APQ and ∆BPQ are congruent | Three sides congruent (sss). PQ is common to both. |
| 5 | Angles APJ, BPJ, AQJ, BQJ are congruent. (The four angles at P and Q with red dots) | CPCTC. Corresponding parts of congruent triangles are congruent |
| Then we prove that the left and right triangles are isosceles and congruent | ||
| 6 | ∆APB and ∆AQB are isosceles | Two sides are congruent (length c) |
| 7 | Angles QAJ, QBJ are congruent. | Base angles of isosceles triangles are congruent |
| 8 | Triangles ∆APB and ∆AQB are congruent | Three sides congruent (sss). AB is common to both. |
| 9 | Angles PAJ, PBJ, QAJ, QBJ are congruent. (The four angles at A and B with blue dots) | CPCTC. Corresponding parts of congruent triangles are congruent |
| Then we prove that the four small triangles are congruent and finish the proof | ||
| 10 | Triangles ∆APJ, ∆BPJ, ∆AQJ, ∆BQJ arecongruent | Two angles and included side (ASA) |
| 11 | The four angles at J - AJP, AJQ, BJP, BJQ arecongruent | CPCTC. Corresponding parts of congruent triangles are congruent |
| 12 | Each of the four angles at J are 90°. Therefore AB is perpendicular to PQ | They are equal in measure and add to 360° |
| 13 | Line segments PJ and QJ are congruent. Therefore AB bisects PQ. | From (8), CPCTC. Corresponding parts of congruent triangles are congruent |
- Q.E.D
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