LIVE 148: 几何工具
10月26日起，每周六上午1:00–2:15 （您所在地当地时间）
该模块直播课总计25小时，课时费约为(~$21/小时)。报名直播课将同时享受该模块对应录播课程（Pre-Recorded Course）一年有效，无限次回看。
课程师生比不低于1:13。老师们将用各种方式鼓励孩子们发言、提问和思考，最大限度地提高孩子们的课堂参与度和积极性。每个班次不超过40名学生。
* 该班次直播课上课标准时间为上午9:00，中国时间。
不知道课程是否合适？来挑战一下能力水平测试吧！
课程日历
上课日由彩色图例标识。点击任意上课日了解当天课程内容
图例
16节日挑战课程（Day X Challenge），通过不同的数学问题学习新的知识内容
4节单元复习课程（Week X Challenge），集中解决单元测验中遇到的问题
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
LIVE在线直播课程每个模块包含【16次日挑战课程（第1课到第16课）+ 4次单元复习课（回顾单元测验Week Challenge 1到4）】共20节直播课程。学生将同时享受该模块对应录播课程（Pre-Recorded Course）一年有效。每四节日挑战直播课结束后将有一次单元考核测验，巩固前4节课学习到的知识内容，帮助学生进一步内化思考。学生需要在复习课前进入Pre-Recorded课程完成单元测验。
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
高中生co-Star老师
SRIRAM KALKI
代表堪萨斯州参加MATHCOUNTS全国赛 • 两次入选AIME资格赛 • USACO（美国计算机奥林匹克）银牌获得者
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YIKAI ZHAO
入围USAMO资格赛 • USACO铂金级选手 • 中国国家省级信息学奥林匹克竞赛一等奖 • 学生会成员 • 喜欢下棋、读书和打网球
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