#### 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？visit our home page first，来看看我们是如何打破线上数学课的固定框架的吧！

#### 点击开启麦克风

### 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。

选择课程前，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【周挑战测验】复习课程（学生需要在复习课前进入配套的Pre-Recorded课程完成周挑战测验）。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。

【周挑战测验】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

This is really helpful. I used to suck at geometry and had little to no interest in it. After I used this course, I had drastic improvements in Geometry. definitely would recommend

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!

Geometry Tools is a great course if you want to learn more about Geometry. Professor Po Shen Loh teaches a lot of great tricks that you can use to solve some difficult problems. I have learned a lot of tricks and tips, including how to square a number, that ends in 5, in just a few seconds. After I completed only 15% of the course, I learned a lot of material that I had not learned in school. I improved greatly on most of my geometry tests. I am very happy about my improvement in Geometry and plan to take the Module 3: Combinatorics Tools and Module 4: Algebra Tools. Thank you, Professor Po Shen Loh, for taking your time to make videos to try to help younger people, and I hope you make more courses!

This comprehensive module helped me a lot with my geometric skills. Highly recommended.

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.

I really like the seminar, it is interesting and challenging; and hard of course. Dr. Po did a great job explaining the question using different approaches.

muy bien

This is really helpful. I used to suck at geometry and had little to no interest in it. After I used this course, I had drastic improvements in Geometry. definitely would recommend

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!

Geometry Tools is a great course if you want to learn more about Geometry. Professor Po Shen Loh teaches a lot of great tricks that you can use to solve some difficult problems. I have learned a lot of tricks and tips, including how to square a number, that ends in 5, in just a few seconds. After I completed only 15% of the course, I learned a lot of material that I had not learned in school. I improved greatly on most of my geometry tests. I am very happy about my improvement in Geometry and plan to take the Module 3: Combinatorics Tools and Module 4: Algebra Tools. Thank you, Professor Po Shen Loh, for taking your time to make videos to try to help younger people, and I hope you make more courses!

This comprehensive module helped me a lot with my geometric skills. Highly recommended.

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.

I really like the seminar, it is interesting and challenging; and hard of course. Dr. Po did a great job explaining the question using different approaches.

muy bien

This is really helpful. I used to suck at geometry and had little to no interest in it. After I used this course, I had drastic improvements in Geometry. definitely would recommend

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!

Geometry Tools is a great course if you want to learn more about Geometry. Professor Po Shen Loh teaches a lot of great tricks that you can use to solve some difficult problems. I have learned a lot of tricks and tips, including how to square a number, that ends in 5, in just a few seconds. After I completed only 15% of the course, I learned a lot of material that I had not learned in school. I improved greatly on most of my geometry tests. I am very happy about my improvement in Geometry and plan to take the Module 3: Combinatorics Tools and Module 4: Algebra Tools. Thank you, Professor Po Shen Loh, for taking your time to make videos to try to help younger people, and I hope you make more courses!

This comprehensive module helped me a lot with my geometric skills. Highly recommended.

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.

I really like the seminar, it is interesting and challenging; and hard of course. Dr. Po did a great job explaining the question using different approaches.

muy bien

This comprehensive module helped me a lot with my geometric skills. Highly recommended.

muy bien

This comprehensive module helped me a lot with my geometric skills. Highly recommended.

muy bien

This comprehensive module helped me a lot with my geometric skills. Highly recommended.

muy bien

This comprehensive module helped me a lot with my geometric skills. Highly recommended.

muy bien

This comprehensive module helped me a lot with my geometric skills. Highly recommended.

muy bien

This comprehensive module helped me a lot with my geometric skills. Highly recommended.

muy bien

This comprehensive module helped me a lot with my geometric skills. Highly recommended.

muy bien

This comprehensive module helped me a lot with my geometric skills. Highly recommended.

muy bien

This comprehensive module helped me a lot with my geometric skills. Highly recommended.

muy bien

This comprehensive module helped me a lot with my geometric skills. Highly recommended.

muy bien

This comprehensive module helped me a lot with my geometric skills. Highly recommended.

muy bien

This comprehensive module helped me a lot with my geometric skills. Highly recommended.

muy bien

This comprehensive module helped me a lot with my geometric skills. Highly recommended.

muy bien

This comprehensive module helped me a lot with my geometric skills. Highly recommended.

muy bien

This comprehensive module helped me a lot with my geometric skills. Highly recommended.

muy bien

This comprehensive module helped me a lot with my geometric skills. Highly recommended.

muy bien

This comprehensive module helped me a lot with my geometric skills. Highly recommended.

muy bien

### 课程大纲

**第1课**

Exterior and interior angles; degrees in a circle; symmetry; sum of angles in triangle; altitude and area of an equilateral triangle; Pythagorean Thm; simplifying radicals; similar triangles and area scaling; 30°-60°-90° triangles; trick for squaring numbers ending in 5

**第2课**

Regular octagons; right isosceles triangles; similar triangles; Pythagorean Thm; simplifying radical denominators; variable equations; polygon area; midpoints; corresponding and alternate interior angles; angle-chasing method

**第3课**

Areas of circular sections; areas of intersections of circles; radius; equilateral triangles; inscribed angles and arcs; area of a triangle; 30°-60°-90° right triangles; area of a lune; ratios of areas of similar figures

**第4课**

Area of parallelograms; altitude of a triangle; special triangles (3-5-7, right, obtuse); exterior angles; manipulating diagrams; 30°-60°-90° triangles; isosceles trapezoids; factoring; variable equations

**第5课**

Right trapezoids; 8-15-17 right triangles; Pythagorean Thm; AAA similarity and scaling; corresponding angles; simplifying ratios; kite area and diagonals; angle chasing; symmetry; complementary angles

**第6课**

Inscribed and central angles; inscribed angle thm; isosceles triangles; supplementary angles; exterior angle thm; variables; arc measure; subtended arcs; secant lines; intersecting secant angle thm; angles of intersecting chords

**第7课**

Chords; midpoints; 8-15-17 and 3-4-5 right triangles; ratios and scaling; inscribed angles and arcs; congruency; Pythagorean Thm; similar triangles; intersecting chords thm; difference of squares; common chords; circumscribed circle on three points

**第8课**

Cyclic quadrilaterals; arc measure; inscribed angles; degrees in a circle; exterior angle thm; supplementary angles; intersecting chord angle thm; counting; secant and tangent lines to a circle; secant-tangent product thm

**第9课**

Concentric circles; right triangles; Pythagorean Thm; symmetry; angle between radius and tangent of a circle; congruent triangles; simplifying radicals; externally tangent circles; equilateral triangles; areas of circular sectors

**第10课**

Angle between tangent and chord to a circle; angle between radius and tangent; intersecting chord angle thm; secant-tangent and tangent-tangent inscribed angles; angles of cyclic quadrilaterals; inscribed angle thm; proof techniques

**第11课**

External and internal tangents to two circles; parallel lines and corresponding angles; Pythagorean Thm; 5-12-13 and 3-4-5 right triangles; similar triangles and scaling; prime numbers; manipulations of diagrams

**第12课**

Tangent-tangent segments to a circle; Pythagorean Thm; kites; similar triangles; angle bisector; area and sums of pairs of sides of quadrilateral with inscribed circle; area of a triangle

**第13课**

Angle bisector thm; isosceles and congruent triangles; area and altitude of a triangle; ratios and scaling; golden ratio; simplifying radical denominators; proof of angle bisector thm; inscribed circle in a triangle; estimation

**第14课**

Reflected angles; minimizing path length; triangle inequality; opposite interior angles; congruency; transversals and corresponding angles; Pythagorean Thm; 3-D path length; manipulating 3-D diagrams; estimation

**第15课**

Triangle inequality; counting; case analysis

**第16课**

Special triangles; area of a triangle; Pythagorean triples; scaling; proof of Heron's formula; semiperimeter; Pythagorean Thm; polynomials; common chord to two circles; 30°-60°-90° triangle

### 报名课程

我们将在2023年夏天开设本模块的直播课程。

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。