几何工具
通过精心编写的题目，快速深化几何概念的理解，即使学生尚未学习高中几何证明课程，也可以轻松学习。该课程难度范围覆盖MATHCOUNTS、AMC 8 和 AMC 10。
想了解我们独具一格的教学方法？欢迎您访问我们的主页，看看我们是如何打破线上数学课的固定框架的吧！
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这门课适合我吗？
你是否正在上从6年级数学到代数的数学课程，并且已经了解综合入门课程中的内容？如果是的话，你已经具备了足够的能力来加入这门课。这门课将带你高效学习如何更具创造力地思考几何，轻松应对比学校课程更难的题目。
不知道课程是否合适？来挑战一下能力水平测试吧！
课程结构
每个模块的课程均包含20节LIVE课程，分为16节【日挑战】课程和4节【单元考核测验】复习课程（学生需要在复习课前进入配套的Pre-Recorded课程完成测验）。每节课程都将由2位 Stars 共同授课。在该模块的学习中，我们将讨论超过150个数学问题。
【日挑战课程】每节课中，学生都会学习解决全新的数学问题，老师们将通过启发式的教学引导和鼓励学生开创自己的解题方法。我们的LIVE课堂赞赏学生们提出自己的想法，鼓励每个学生和全班同学一起集思广益，构建各种不同的解题方法，更积极地参与数学发现。
【单元考核测验】包含20道数学问题，其中10题在正常难度范围内，而另外10题则非常具有挑战性。
学生评价
Before I took this course, geometry was a part of math that I was okay at, but didn't consider one of my top aspects of mathematics. As I took this course, I began to discover the other side of geometry, and how it was in fact a very important part of math. Now, I consider geometry as one of my favorite parts of math to work with, thanks to this course!
我在第三次尝试时完成作业时拿到了满分，但我没能解出附加题和超级附加题。不过，在这次复习课后，很多问题思路都开始变得清晰起来，虽然有些问题有点让人困惑，但最终我搞明白了
This comprehensive module helped me a lot with my geometric skills. Highly recommended.
我真的会想念这堂课。非常感谢。
Very impressive, provides many ways to solve problems. The problems are built like riddles, very clever and they always have clear solutions that are properly explained.
和你一起上课真的很有趣💕
This is a very fun (but hard!) course. I like how the problems are given to introduce new concepts and the solutions are very clear, and how even though there's a lot of stuff covered, it's presented in only a few problems.
我学到了很多🙂
Before I took this course, geometry was a part of math that I was okay at, but didn't consider one of my top aspects of mathematics. As I took this course, I began to discover the other side of geometry, and how it was in fact a very important part of math. Now, I consider geometry as one of my favorite parts of math to work with, thanks to this course!
我在第三次尝试时完成作业时拿到了满分，但我没能解出附加题和超级附加题。不过，在这次复习课后，很多问题思路都开始变得清晰起来，虽然有些问题有点让人困惑，但最终我搞明白了
This comprehensive module helped me a lot with my geometric skills. Highly recommended.
我真的会想念这堂课。非常感谢。
Very impressive, provides many ways to solve problems. The problems are built like riddles, very clever and they always have clear solutions that are properly explained.
和你一起上课真的很有趣💕
This is a very fun (but hard!) course. I like how the problems are given to introduce new concepts and the solutions are very clear, and how even though there's a lot of stuff covered, it's presented in only a few problems.
我学到了很多🙂
Before I took this course, geometry was a part of math that I was okay at, but didn't consider one of my top aspects of mathematics. As I took this course, I began to discover the other side of geometry, and how it was in fact a very important part of math. Now, I consider geometry as one of my favorite parts of math to work with, thanks to this course!
我在第三次尝试时完成作业时拿到了满分，但我没能解出附加题和超级附加题。不过，在这次复习课后，很多问题思路都开始变得清晰起来，虽然有些问题有点让人困惑，但最终我搞明白了
This comprehensive module helped me a lot with my geometric skills. Highly recommended.
我真的会想念这堂课。非常感谢。
Very impressive, provides many ways to solve problems. The problems are built like riddles, very clever and they always have clear solutions that are properly explained.
和你一起上课真的很有趣💕
This is a very fun (but hard!) course. I like how the problems are given to introduce new concepts and the solutions are very clear, and how even though there's a lot of stuff covered, it's presented in only a few problems.
我学到了很多🙂
Before I took this course, geometry was a part of math that I was okay at, but didn't consider one of my top aspects of mathematics. As I took this course, I began to discover the other side of geometry, and how it was in fact a very important part of math. Now, I consider geometry as one of my favorite parts of math to work with, thanks to this course!
我在第三次尝试时完成作业时拿到了满分，但我没能解出附加题和超级附加题。不过，在这次复习课后，很多问题思路都开始变得清晰起来，虽然有些问题有点让人困惑，但最终我搞明白了
This comprehensive module helped me a lot with my geometric skills. Highly recommended.
我真的会想念这堂课。非常感谢。
Very impressive, provides many ways to solve problems. The problems are built like riddles, very clever and they always have clear solutions that are properly explained.
和你一起上课真的很有趣💕
This is a very fun (but hard!) course. I like how the problems are given to introduce new concepts and the solutions are very clear, and how even though there's a lot of stuff covered, it's presented in only a few problems.
我学到了很多🙂
Before I took this course, geometry was a part of math that I was okay at, but didn't consider one of my top aspects of mathematics. As I took this course, I began to discover the other side of geometry, and how it was in fact a very important part of math. Now, I consider geometry as one of my favorite parts of math to work with, thanks to this course!
我在第三次尝试时完成作业时拿到了满分，但我没能解出附加题和超级附加题。不过，在这次复习课后，很多问题思路都开始变得清晰起来，虽然有些问题有点让人困惑，但最终我搞明白了
This comprehensive module helped me a lot with my geometric skills. Highly recommended.
我真的会想念这堂课。非常感谢。
Very impressive, provides many ways to solve problems. The problems are built like riddles, very clever and they always have clear solutions that are properly explained.
和你一起上课真的很有趣💕
This is a very fun (but hard!) course. I like how the problems are given to introduce new concepts and the solutions are very clear, and how even though there's a lot of stuff covered, it's presented in only a few problems.
我学到了很多🙂
Before I took this course, geometry was a part of math that I was okay at, but didn't consider one of my top aspects of mathematics. As I took this course, I began to discover the other side of geometry, and how it was in fact a very important part of math. Now, I consider geometry as one of my favorite parts of math to work with, thanks to this course!
我在第三次尝试时完成作业时拿到了满分，但我没能解出附加题和超级附加题。不过，在这次复习课后，很多问题思路都开始变得清晰起来，虽然有些问题有点让人困惑，但最终我搞明白了
This comprehensive module helped me a lot with my geometric skills. Highly recommended.
我真的会想念这堂课。非常感谢。
Very impressive, provides many ways to solve problems. The problems are built like riddles, very clever and they always have clear solutions that are properly explained.
和你一起上课真的很有趣💕
This is a very fun (but hard!) course. I like how the problems are given to introduce new concepts and the solutions are very clear, and how even though there's a lot of stuff covered, it's presented in only a few problems.
我学到了很多🙂
Before I took this course, geometry was a part of math that I was okay at, but didn't consider one of my top aspects of mathematics. As I took this course, I began to discover the other side of geometry, and how it was in fact a very important part of math. Now, I consider geometry as one of my favorite parts of math to work with, thanks to this course!
我在第三次尝试时完成作业时拿到了满分，但我没能解出附加题和超级附加题。不过，在这次复习课后，很多问题思路都开始变得清晰起来，虽然有些问题有点让人困惑，但最终我搞明白了
This comprehensive module helped me a lot with my geometric skills. Highly recommended.
我真的会想念这堂课。非常感谢。
Very impressive, provides many ways to solve problems. The problems are built like riddles, very clever and they always have clear solutions that are properly explained.
和你一起上课真的很有趣💕
This is a very fun (but hard!) course. I like how the problems are given to introduce new concepts and the solutions are very clear, and how even though there's a lot of stuff covered, it's presented in only a few problems.
我学到了很多🙂
Before I took this course, geometry was a part of math that I was okay at, but didn't consider one of my top aspects of mathematics. As I took this course, I began to discover the other side of geometry, and how it was in fact a very important part of math. Now, I consider geometry as one of my favorite parts of math to work with, thanks to this course!
我在第三次尝试时完成作业时拿到了满分，但我没能解出附加题和超级附加题。不过，在这次复习课后，很多问题思路都开始变得清晰起来，虽然有些问题有点让人困惑，但最终我搞明白了
This comprehensive module helped me a lot with my geometric skills. Highly recommended.
我真的会想念这堂课。非常感谢。
Very impressive, provides many ways to solve problems. The problems are built like riddles, very clever and they always have clear solutions that are properly explained.
和你一起上课真的很有趣💕
This is a very fun (but hard!) course. I like how the problems are given to introduce new concepts and the solutions are very clear, and how even though there's a lot of stuff covered, it's presented in only a few problems.
我学到了很多🙂
Before I took this course, geometry was a part of math that I was okay at, but didn't consider one of my top aspects of mathematics. As I took this course, I began to discover the other side of geometry, and how it was in fact a very important part of math. Now, I consider geometry as one of my favorite parts of math to work with, thanks to this course!
我在第三次尝试时完成作业时拿到了满分，但我没能解出附加题和超级附加题。不过，在这次复习课后，很多问题思路都开始变得清晰起来，虽然有些问题有点让人困惑，但最终我搞明白了
This comprehensive module helped me a lot with my geometric skills. Highly recommended.
我真的会想念这堂课。非常感谢。
Very impressive, provides many ways to solve problems. The problems are built like riddles, very clever and they always have clear solutions that are properly explained.
和你一起上课真的很有趣💕
This is a very fun (but hard!) course. I like how the problems are given to introduce new concepts and the solutions are very clear, and how even though there's a lot of stuff covered, it's presented in only a few problems.
我学到了很多🙂
Before I took this course, geometry was a part of math that I was okay at, but didn't consider one of my top aspects of mathematics. As I took this course, I began to discover the other side of geometry, and how it was in fact a very important part of math. Now, I consider geometry as one of my favorite parts of math to work with, thanks to this course!
我在第三次尝试时完成作业时拿到了满分，但我没能解出附加题和超级附加题。不过，在这次复习课后，很多问题思路都开始变得清晰起来，虽然有些问题有点让人困惑，但最终我搞明白了
This comprehensive module helped me a lot with my geometric skills. Highly recommended.
我真的会想念这堂课。非常感谢。
Very impressive, provides many ways to solve problems. The problems are built like riddles, very clever and they always have clear solutions that are properly explained.
和你一起上课真的很有趣💕
This is a very fun (but hard!) course. I like how the problems are given to introduce new concepts and the solutions are very clear, and how even though there's a lot of stuff covered, it's presented in only a few problems.
我学到了很多🙂
Before I took this course, geometry was a part of math that I was okay at, but didn't consider one of my top aspects of mathematics. As I took this course, I began to discover the other side of geometry, and how it was in fact a very important part of math. Now, I consider geometry as one of my favorite parts of math to work with, thanks to this course!
我在第三次尝试时完成作业时拿到了满分，但我没能解出附加题和超级附加题。不过，在这次复习课后，很多问题思路都开始变得清晰起来，虽然有些问题有点让人困惑，但最终我搞明白了
This comprehensive module helped me a lot with my geometric skills. Highly recommended.
我真的会想念这堂课。非常感谢。
Very impressive, provides many ways to solve problems. The problems are built like riddles, very clever and they always have clear solutions that are properly explained.
和你一起上课真的很有趣💕
This is a very fun (but hard!) course. I like how the problems are given to introduce new concepts and the solutions are very clear, and how even though there's a lot of stuff covered, it's presented in only a few problems.
我学到了很多🙂
Before I took this course, geometry was a part of math that I was okay at, but didn't consider one of my top aspects of mathematics. As I took this course, I began to discover the other side of geometry, and how it was in fact a very important part of math. Now, I consider geometry as one of my favorite parts of math to work with, thanks to this course!
我在第三次尝试时完成作业时拿到了满分，但我没能解出附加题和超级附加题。不过，在这次复习课后，很多问题思路都开始变得清晰起来，虽然有些问题有点让人困惑，但最终我搞明白了
This comprehensive module helped me a lot with my geometric skills. Highly recommended.
我真的会想念这堂课。非常感谢。
Very impressive, provides many ways to solve problems. The problems are built like riddles, very clever and they always have clear solutions that are properly explained.
和你一起上课真的很有趣💕
This is a very fun (but hard!) course. I like how the problems are given to introduce new concepts and the solutions are very clear, and how even though there's a lot of stuff covered, it's presented in only a few problems.
我学到了很多🙂
Before I took this course, geometry was a part of math that I was okay at, but didn't consider one of my top aspects of mathematics. As I took this course, I began to discover the other side of geometry, and how it was in fact a very important part of math. Now, I consider geometry as one of my favorite parts of math to work with, thanks to this course!
我在第三次尝试时完成作业时拿到了满分，但我没能解出附加题和超级附加题。不过，在这次复习课后，很多问题思路都开始变得清晰起来，虽然有些问题有点让人困惑，但最终我搞明白了
This comprehensive module helped me a lot with my geometric skills. Highly recommended.
我真的会想念这堂课。非常感谢。
Very impressive, provides many ways to solve problems. The problems are built like riddles, very clever and they always have clear solutions that are properly explained.
和你一起上课真的很有趣💕
This is a very fun (but hard!) course. I like how the problems are given to introduce new concepts and the solutions are very clear, and how even though there's a lot of stuff covered, it's presented in only a few problems.
我学到了很多🙂
Before I took this course, geometry was a part of math that I was okay at, but didn't consider one of my top aspects of mathematics. As I took this course, I began to discover the other side of geometry, and how it was in fact a very important part of math. Now, I consider geometry as one of my favorite parts of math to work with, thanks to this course!
我在第三次尝试时完成作业时拿到了满分，但我没能解出附加题和超级附加题。不过，在这次复习课后，很多问题思路都开始变得清晰起来，虽然有些问题有点让人困惑，但最终我搞明白了
This comprehensive module helped me a lot with my geometric skills. Highly recommended.
我真的会想念这堂课。非常感谢。
Very impressive, provides many ways to solve problems. The problems are built like riddles, very clever and they always have clear solutions that are properly explained.
和你一起上课真的很有趣💕
This is a very fun (but hard!) course. I like how the problems are given to introduce new concepts and the solutions are very clear, and how even though there's a lot of stuff covered, it's presented in only a few problems.
我学到了很多🙂
Before I took this course, geometry was a part of math that I was okay at, but didn't consider one of my top aspects of mathematics. As I took this course, I began to discover the other side of geometry, and how it was in fact a very important part of math. Now, I consider geometry as one of my favorite parts of math to work with, thanks to this course!
我在第三次尝试时完成作业时拿到了满分，但我没能解出附加题和超级附加题。不过，在这次复习课后，很多问题思路都开始变得清晰起来，虽然有些问题有点让人困惑，但最终我搞明白了
This comprehensive module helped me a lot with my geometric skills. Highly recommended.
我真的会想念这堂课。非常感谢。
Very impressive, provides many ways to solve problems. The problems are built like riddles, very clever and they always have clear solutions that are properly explained.
和你一起上课真的很有趣💕
This is a very fun (but hard!) course. I like how the problems are given to introduce new concepts and the solutions are very clear, and how even though there's a lot of stuff covered, it's presented in only a few problems.
我学到了很多🙂
Before I took this course, geometry was a part of math that I was okay at, but didn't consider one of my top aspects of mathematics. As I took this course, I began to discover the other side of geometry, and how it was in fact a very important part of math. Now, I consider geometry as one of my favorite parts of math to work with, thanks to this course!
我在第三次尝试时完成作业时拿到了满分，但我没能解出附加题和超级附加题。不过，在这次复习课后，很多问题思路都开始变得清晰起来，虽然有些问题有点让人困惑，但最终我搞明白了
This comprehensive module helped me a lot with my geometric skills. Highly recommended.
我真的会想念这堂课。非常感谢。
Very impressive, provides many ways to solve problems. The problems are built like riddles, very clever and they always have clear solutions that are properly explained.
和你一起上课真的很有趣💕
This is a very fun (but hard!) course. I like how the problems are given to introduce new concepts and the solutions are very clear, and how even though there's a lot of stuff covered, it's presented in only a few problems.
我学到了很多🙂
Before I took this course, geometry was a part of math that I was okay at, but didn't consider one of my top aspects of mathematics. As I took this course, I began to discover the other side of geometry, and how it was in fact a very important part of math. Now, I consider geometry as one of my favorite parts of math to work with, thanks to this course!
我在第三次尝试时完成作业时拿到了满分，但我没能解出附加题和超级附加题。不过，在这次复习课后，很多问题思路都开始变得清晰起来，虽然有些问题有点让人困惑，但最终我搞明白了
This comprehensive module helped me a lot with my geometric skills. Highly recommended.
我真的会想念这堂课。非常感谢。
Very impressive, provides many ways to solve problems. The problems are built like riddles, very clever and they always have clear solutions that are properly explained.
和你一起上课真的很有趣💕
This is a very fun (but hard!) course. I like how the problems are given to introduce new concepts and the solutions are very clear, and how even though there's a lot of stuff covered, it's presented in only a few problems.
我学到了很多🙂
Before I took this course, geometry was a part of math that I was okay at, but didn't consider one of my top aspects of mathematics. As I took this course, I began to discover the other side of geometry, and how it was in fact a very important part of math. Now, I consider geometry as one of my favorite parts of math to work with, thanks to this course!
我在第三次尝试时完成作业时拿到了满分，但我没能解出附加题和超级附加题。不过，在这次复习课后，很多问题思路都开始变得清晰起来，虽然有些问题有点让人困惑，但最终我搞明白了
This comprehensive module helped me a lot with my geometric skills. Highly recommended.
我真的会想念这堂课。非常感谢。
Very impressive, provides many ways to solve problems. The problems are built like riddles, very clever and they always have clear solutions that are properly explained.
和你一起上课真的很有趣💕
This is a very fun (but hard!) course. I like how the problems are given to introduce new concepts and the solutions are very clear, and how even though there's a lot of stuff covered, it's presented in only a few problems.
我学到了很多🙂
Before I took this course, geometry was a part of math that I was okay at, but didn't consider one of my top aspects of mathematics. As I took this course, I began to discover the other side of geometry, and how it was in fact a very important part of math. Now, I consider geometry as one of my favorite parts of math to work with, thanks to this course!
我在第三次尝试时完成作业时拿到了满分，但我没能解出附加题和超级附加题。不过，在这次复习课后，很多问题思路都开始变得清晰起来，虽然有些问题有点让人困惑，但最终我搞明白了
This comprehensive module helped me a lot with my geometric skills. Highly recommended.
我真的会想念这堂课。非常感谢。
Very impressive, provides many ways to solve problems. The problems are built like riddles, very clever and they always have clear solutions that are properly explained.
和你一起上课真的很有趣💕
This is a very fun (but hard!) course. I like how the problems are given to introduce new concepts and the solutions are very clear, and how even though there's a lot of stuff covered, it's presented in only a few problems.
我学到了很多🙂
Before I took this course, geometry was a part of math that I was okay at, but didn't consider one of my top aspects of mathematics. As I took this course, I began to discover the other side of geometry, and how it was in fact a very important part of math. Now, I consider geometry as one of my favorite parts of math to work with, thanks to this course!
我在第三次尝试时完成作业时拿到了满分，但我没能解出附加题和超级附加题。不过，在这次复习课后，很多问题思路都开始变得清晰起来，虽然有些问题有点让人困惑，但最终我搞明白了
This comprehensive module helped me a lot with my geometric skills. Highly recommended.
我真的会想念这堂课。非常感谢。
Very impressive, provides many ways to solve problems. The problems are built like riddles, very clever and they always have clear solutions that are properly explained.
和你一起上课真的很有趣💕
This is a very fun (but hard!) course. I like how the problems are given to introduce new concepts and the solutions are very clear, and how even though there's a lot of stuff covered, it's presented in only a few problems.
我学到了很多🙂
课程大纲
第1课
内⻆和外⻆、圆的度数、对称、三⻆形内⻆和、等边三⻆形的高和面积、勾股定理、根式的简化、相似三⻆形及其面积比例、30°-60°-90°三⻆形、以5结尾的数字平方技巧
第2课
正八边形、等腰直⻆三⻆形、相似三⻆形、勾股定理、根式分母的简化、方程组、多边形面积、中点、平行线同位⻆及内错⻆性质、追⻆法
第3课
圆形截面面积、圆的重叠部分面积、半径、等边三⻆形、圆周⻆的⻆度及其对应弧⻓、三⻆形面积、30°-60°-90°直⻆三⻆形、月牙型面积、相似图形的面积比
第4课
平行四边形面积、三⻆形的高、特殊三⻆形(3-5-7边⻓三⻆形、直⻆三⻆形、钝⻆三⻆形)、外⻆、图形变换、30°-60°-90°三⻆形、等腰梯形、因式分解、方程组
第5课
直⻆梯形、 8-15-17边⻓直⻆三⻆形、勾股定理、⻆-⻆-⻆相似三⻆形及比例、同位⻆、简化比例、筝形面积和对⻆线性质、追⻆法、对称、余⻆
第6课
圆周⻆和圆心⻆、圆周⻆定理、等腰三⻆形、补⻆、外⻆定理、变量、弧度、任意⻆所对应的圆弧、割线、切割线定理、相交弦所成⻆度
第7课
弦、中点、8-15-17边⻓直⻆三⻆形、3-4-5边⻓直⻆三⻆形、边⻓和面积比、圆周⻆和弧度、全等图形、勾股定理、相似三⻆形、相交弦定理、平方差、两圆公共弦、三⻆形外接圆
第8课
圆内接四边形、弧度、圆周⻆、圆的度数、外⻆定理、补⻆、相交弦所成⻆度定理、计数、圆的割线和切线、割切线定理
第9课
同心圆、直⻆三⻆形、勾股定理、对称、圆的切线及过其切点的半径所成⻆度、全等三⻆形、根式的简化、外切圆、等边三⻆形、扇形面积
第10课
切线和弦的夹⻆、圆的切线及过其切点的半径所成⻆度、相交弦⻆定理、切割线内⻆、切线-切线内⻆、圆内接四边形⻆度、圆周⻆定理、证明技巧
第11课
两圆内外切线、平行线和同位⻆、勾股定理、5-12-13边⻓和3-4-5边⻓直⻆三⻆形、相似三⻆形及其比例、质数、图形变换
第12课
过圆外一点的切线线段、勾股定理、筝形、相似三⻆形、⻆平分线、带内切圆的四边形的面积和数量、三⻆形面积
第13课
⻆平分线定理、等腰全等三⻆形、三⻆形的面积和高、比、⻩金比例、根式分母的简化、⻆平分线定理的证明、三⻆形的内接圆、估算
第14课
反射⻆、最短路径、三⻆形不等式、内对⻆、全等、截线和同位⻆、勾股定理、3-D路径⻓度、3-D图形处理、估算
第15课
三⻆形不等式、计数、案例分析
第16课
特殊三⻆形、三⻆形的面积、毕氏三数、比例、海伦公式证明、半周⻓、勾股定理、多项式、两圆 公共弦、30°-60°-90°⻆度三⻆形
报名课程
每位报名LIVE在线直播课的学生都将同时享有对应的Pre-Recorded自学录播课程一年使用权，无需额外购买。LIVE直播课经过罗教授和教学团队精心设计的，通过前所未有的引人入胜的教学方式、极为丰富的课堂交互体验，让学生在趣味学习中爱上挑战和思考。 以下所有直播课显示时间均为您的当地时间（美国将于2023年3月12日星期日进入夏令时，请其他国家和地区的学生注意课程时间调整）。
如果你有强大的自驱力可以进行自主学习，或者想比LIVE课程安排更快地完成一个模块的学习，也欢迎你选择我们的自学录播课程。录播课包含了所有罗教授关于该模块的教学视频（没有直播课堂），在过去3年中也有数千名来自世界各地的同学参与学习。