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Brilliant

Geometry II

via Brilliant

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Overview

This course covers a wide range of theorems in classical Euclidean geometry. You'll start by deriving the Central Angle Theorem and Thales' Theorem, then move on to the Power of a Point Theorem, and conclude with an exploration of different types of triangle centers and their presence on the Euler Line.

Our goal is to help you understand and explore the derivations of these theorems, and to give you many opportunities to practice and strengthen your skills applying them!
Algebra is used throughout this course, but nothing beyond the level of Algebra I. This course is a great place to start (or continue) if you're already familiar with the content in Geometry I.

Syllabus

  • Introduction: Circles form a central part of geometry: get started with your first theorems!
    • Central Angles and Arcs: Investigate geometric patterns and proofs that utilize the angles at the center of circles.
    • Thales's Theorem: What happens when an angle is inscribed in a semicircle?
    • Inscribed Angles: Extend Thales' observation into another beautiful and more general theorem.
  • In and Out of Circles: Shapes and angles inscribed and circumscribed.
    • Puzzles with Inscribed Angles: Warm up with this round of practice problems that explore the Inscribed Angle Theorem.
    • Cyclic Quadrilaterals: Study the properties of quadrilaterals inscribed inside of circles.
    • Power of a Point I: Intersecting lines inside a circle are a special circumstance worth investigating in detail!
    • Intersecting Secants: What happens when lines intersect inside a circle?
    • Power of a Point II: Solve a problem assortment that puts all of your new theorems to use!
    • Tangents: What happens when lines just barely touch the outside of a circle?
    • Problem-Solving Challenges: Practice and strengthen the entire set of tools you've learned so far with this final round of challenges.
  • Mastering Triangles: Master the inner secrets of triangles.
    • Right Triangles: Start your journey into advanced triangles on the right (aka 90-degree) foot.
    • Thales + Pythagoras: Combine what you know about Thales and Pythagoras to approach some fascinating problems.
    • Cevians: Explore what happens when you connect up one point and one side.
    • Pegboard Triangles: What happens when the triangles are drawn on a regular grid?
  • Triangle Centers: The Euler Line will blow your mind.
    • Three Different Centers: Learn about the three most commonly used triangle centers and explore how they relate to each other.
    • The Circumcenter: Use perpendicular bisectors to experiment with a fourth type of "center."
    • Euler's Line: Prove a profound result that relates the orthocenter, centroid, and circumcenter.
    • Morley's Triangle: Use trisectors rather than bisectors to get an astonishing result.

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