3-4-5 三角形问题The 3-4-5 Triangle Problem

20° The 3-4-5 triangle problem. There are reports that ancient Egyptian surveyors laid out right angles by constructing 3–4–5 triangles with a rope divided into 12 equal parts by 11 knots. Since there is no documentary evidence to the effect that the Egyptians were aware of even a particular case of the Pythagorean theorem, the following purely academic problem arises: Show, without using the Pythagorean theorem, its converse, or any of its consequences, that the 3–4–5 triangle is a right triangle. The reader may care to try to solve this problem. Many different solutions have been given over the years. A particularly simple solution can be effected based on Figure 3, which shows four right triangles with legs of 3 and 4 units, along with a unit square, making up a square of 25 square units of area. This means that the hypotenuse of a right triangle with legs of 3 and 4 units is 5 units long, and it follows that a 3–4–5 triangle is a right triangle. This solution is found in the Chou-pei Suan-king (The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven), a work generally regarded as the oldest of the ancient Chinese mathematical classics. The solution is easily generalized to obtain the Pythagorean relation for an arbitrary right triangle.