本书集周春荔教授毕生所学，将几何辅助线的添加方法和原理娓娓道来，充分体现"数学是智力的磨刀石，对于所有信奉教育的人而言，是一种不可缺少的思维训练”的育人作用。几何定理的证明，除少数简易的以外，非添加有用的辅助线，否则就无从着手。辅助线的作法，千变万化，没有一定的方法可以遵循，所以是证题时最困难的一件事。在普通几何书中，很少有将几何辅助线的原理、方法和建构讲得如此清晰明了，学生系统学习后，会得心应手解决几何相关问题。

著名数学教育家许莼舫先生（1906—1965）在《几何定理和证题》一书中曾写道：“几何定理的证明，除少数简单的题目外，都需要添加辅助线. 辅助线的作法千变万化，没有一定的方法可以遵循，所以如何作辅助线是证明几何题时最困难的一件事. 由于无从谈起作辅助线的方法，所以很多几何书中宁可不说也不肯乱说，这使学生感觉十分头痛.”许老先生在书中为初学几何者叙述了一些作辅助线的大要，提纲挈领地提出了10 点. 许老先生的提示，使作者从初中学几何时就不断思索并积累添加辅助线的方法. 与“教学有法，教无定法”类似，添加辅助线总体有法（规律），但对具体问题又无定法，会因人而异、因题而异.数学老师在教学中总结了各种添加辅助线的模型，我们要进一步探索这些模型背后的图形变换原理与思维方式. “会当凌绝顶，一览众山小”，只有对规律有更深层次的理解，才能对几何证题和神奇的辅助线有更深刻的认识. 其实，学任何一门学科都要刻苦、认真，勤于思索，善于总结. 找到了学科的本质、方法的精髓，熟能生巧，最后总能够融会贯通，有所发现，有所创新，学习平面几何也是如此.不会解几何题，根源并不在不会添加辅助线，要害在于对解题思维的基本理论知识和基本方法不够了解，或理解不够深刻. 因此，为了帮助大家学习如何添加辅助线，第一章我们先讲“几何证题知识概述”，简要介绍命题的四种形式与充分必要条件，解数学题的分析思路和几种探索证明途径. 按分析思路逐步实行，就可以顺利地解题成功. 而添加辅助线只是在展开思路遇阻时起到逢山开路，遇水架桥的作用，应是应运而生的，涉及的方法包括平面几何的初等变换（本书只限合同变换、位似变换、等积变形），分合割补，运动变化等. 这些方法神奇的几何辅助线以简驭繁，能够像七巧板拼图那样变化出多彩奇妙的几何图形. 第二章“神奇的几何辅助线”，我们先通过简单的例题说明各种初等几何变换对添加辅助线的作用，然后通过讲解典型的几何问题，洞悉前人思考添加辅助线的秘密. 有诗为证：图形割补勾股弦，几何变换简驭繁，折叠剪拼难化易，学好规矩绘方圆.从操作角度分析，添加辅助线是在构造图形，从思维角度看是数学的“建构思维”，将建构思维中通过添加辅助线解题的方法进一步升华，就是构造图形解题. 这就是第三章“构造图形解题漫谈”中要介绍的内容. 以上只是作者学习几何的一点粗浅体会，愿与读者交流共勉，希望能对几何爱好者有所助益，并期待与大家进一步深入展开对几何解题添加辅助线问题的探索.

第一章 几何证题知识概述 ······························································.1 1.1 命题的四种形式与充分必要条件 ············································.2 1.1.1 命题的四种形式 ························································.2 1.1.2 充分条件与必要条件 ···················································.4 1.2 分析数学题的思路 ·····························································.11 1.2.1 倒推分析思路 ··························································.11 1.2.2 分析综合思路 ··························································.13 1.2.3 反设分析思路 ··························································.15 1.3 几种数学解题探索方法 ·······················································.17 1.3.1 试验发现法 ·····························································.17 1.3.2 联想类比法 ·····························································.18 1.3.3 反例证伪法 ·····························································.19 1.3.4 数形结合的构造图形法 ···············································.19 第二章 神奇的几何辅助线 ·····························································.25 2.1 添加辅助线的目的 ·····························································.25 2.2 添加辅助线的原则 ·····························································.30 原则一 化繁为简 ·····························································.30 原则二 相对集中 ·····························································.31 原则三 作图构造 ·····························································.33 原则四 显现特殊性 ··························································.34 2.3 名题剖析智慧精华 ·····························································.35 2.4 图形变换与辅助线 ·····························································.38 2.4.1 应用平移变换添加辅助线 ············································.39 2.4.2 反射变换添加辅助线作法 ············································.50 2.4.3 通过旋转变换添加辅助线的作用 ···································.63 2.4.4 利用中心对称添加辅助线 ············································.73 2.4.5 利用相似变换添加辅助线 ············································.76 2.4.6 利用等积变换添加辅助线 ············································.78 2.4.7 构造圆进行添加辅助线 ···············································.80 2.4.8 利用复合变换添加辅助线 ············································.88 2.5 综合示范添加辅助线解题 ····················································.90 2.5.1 经典例题添加辅助线赏析 ············································.91 2.5.2 著名竞赛题添加辅助线选析 ·······································.109 2.5.3 一题多证添加辅助线例谈 ··········································.132 第三章 例说构造图形解题 ···························································.141 3.1 构造图形解几何题 ···························································.143 3.2 构造图形解三角题 ···························································.153 3.3 构造图形解代数题 ···························································.162