A circle is the set of points in a plane that are equidistant from a given point. We know that circles have many geometric properties on plane. However, when we use the view in 3D to discuss circles, we may find something interesting in it. In the thesis, I have interest in the inverse circular discs of Platonic solids.

First of all, we will mention some basic objects and criterions, the most important of them is Desargues' Theorem. We have to use it to find the collinear line of the combination of the circular discs. Next, we will show how to use the inversion to construct these circular discs, and then we need to find the common properties of them.

We use Cabri 3D to display five common properteis and constuction, all of graphs and dynamic geometric files show clearly in the following website: http://apollonius.math.nthu.edu.tw/d1/dg-09exe/621610/Thesis/Paper/index.htm

我們在平面上任取一點，圓就是一種收集所有與此點等距離的集合。我們知道圓在平面上有許多幾何性質。不過當我們用立體的視角去觀察圓的時候，我們也許可以找到一些有趣的東西。而在這篇論文中，我對五種柏拉圖立體內的反演內切圓特別感興趣。

在這篇論文中，我們會先提到一些基本的物件與定理，其中最重要的就是Desargues定理。我們必須使用這個定理去找這些圓盤的共線。接下來我們會說明怎麼用反演的方法製作這些圓盤，然後找出公切線的共點性質。

我們會用Cabri 3D展示五種共通的性質以及建構過程，所有圖片和影像在http://apollonius.math.nthu.edu.tw/d1/dg-09exe/621610/Thesis/Paper/index.htm裡都可以清楚看到。

Content
1 Introduction 1
2 Basic criterion 1
3 Construction of circular disc 5
4 The Properties of the Circular Disc in 3D 9
4-1 Four Inverse Circular Discs of Regular Tetrahedron 10
4-2 Six Inverse Circular Discs of Cube 16
4-3 Eight Inverse Circular Discs of Regular Octahedron 25
4-4 Twelve Inverse Circular Discs of Regular Dodecahedron 30
4-5 Twenty Inverse Circular Discs of Regular Icosahedron 38
5 References 43

[1] David A. Brannan, Matthew F. Esplen, Jeremy J. Gray, "Geometry", 2003, Cambridge University Press.
[2] D. Hilbert, S. Cohn-Vossen, "Geometry and the Imagination", 1932, Chelsea.
[3] James R. Smart, “Modern Geometries”, 1994, Brooks.
[4] Roger A. Johnson, “Advanced Euclidean Geometry”, 1929, Dover.
[5] 嚴鎮軍, "反射與反演", 2002, 九章出版社.
[6] http://140.114.32.1/~jcchuan/
[7] http://mathworld.wolfram.com/Circle.html
[8] http://en.wikipedia.org/wiki/Platonic_solid
[9] http://en.wikipedia.org/wiki/Desargues'_theorem
[10] http://mathworld.wolfram.com/DesarguesTheorem.html
[11] http://mathworld.wolfram.com/Concurrent.html
[12] http://mathworld.wolfram.com/Collinear.html