Look for and Make Use of Structure, Symmetry, and Similarity

Here we provide a progression of geometry problems which show a collection of mathematical strategies.  Our goal is always to help students see connections in mathematics and help students leverage connections to solve problems.

The progression of geometry problems here show how looking for and making use of

  • Structure,
  • Symmetry,
  • Similarity, and
  • Area

can help one solve the problem.

These problems regularly use algebra (algebraic reasoning) and proportional reasoning. Many use area. The rule of thumb needs to be:

In Geometry, expect to use Algebra.

The problems in the progression have the added feature that many involve the Geometric Mean or the Harmonic Mean. Geometric Mean and the Harmonic Mean are useful math topics in and of themselves. 

This progression of problems assumes the student has some background in high school geometry — roughly one semester of high school geometry.

The problems begin with problems which are typically found in high school geometry textbooks and move into more problems which may or may not appear in high school textbooks.

This progression of problems is intended for:

  • High School Geometry students.
  • High School students who are preparing for mathematics competitions such as
    • AMC 10 (American Mathematics Competition for 10th grade and under students)
    • AMC 12
    • State mathematics competitions
  • University students majoring in Mathematics Teacher Education (prospective teachers).
  • Anyone interested in seeing the elegant interconnectedness and beauty of mathematics.