摘要或動機

「重複圖形」是本篇報告研究的問題，我們利用「方程式」建立一個尋找重複圖形，並証明其個數的方法。利用此方法得出下面的結論：1.會形成lap 2 的凸多邊形只有2
種，即三角形和四邊形。(1)「lap 2 三角形」只有1 種，即等腰直角三角形。(2)「lap 2 四邊形」只有1 種，即二邊之比為1：
且內角是45°、135°的平行四邊形。2.會形成lap
3 的凸多邊形只有2 種，即三角形和四邊形。(1)「lap 3 三角形」只有1 種，即內角為30°–60°–90°的直角三角形。3.其他的lap k 三角形：(1)任意內角為30°–60°–90°的直角三角形都是lap 3k²，其中k是正整數。(2)邊長比為1：m：
的直角三角形是lap (m²+1)k²三角形，其中m、k是正整數。

To find repeated figures, we construct a method to search them with the help of
algebraic equations. Here we arrive at：1. There are only two kinds of lap 2 convex
polygons, triangles and quadrilaterals. (1) The only lap 2 triangle is isogonal
right-angled. (2) The only lap 2 quadrilateral is the one that contains angles 45°,
90° and two neighboring sides with the ratio 1:

. 2. There are also two kinds
of lap 3 convex polygons, triangles and quadrilaterals. (1) The only lap 3 triangle
is the one with angles 30°, 60° and 90°. 3. Other kinds of lap k triangles are listed
as following: (1) A triangle with angles 30o, 60°, 90° is a lap 3k², the k is a
natural number. (2) A right-angled triangle whose ratio is 1 : m :


is a lap
(m2+1)k², the m and the k are natural numbers.