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    • ▾High school geometry
      • ▸Basics of geometry
        • •Proofs in this course
        • •Geometric definitions
        • •Linear pair theorem
        • •Area of a regular polygon
        • •Diagonal length of a right rectangular prism
        • •Distance and midpoint in 3D
        • •Circumcircles and incircles
      • ▸Transformations
        • •Preserved properties of rigid transformations
        • •Find measures using rigid transformations
        • •Find the angle of rotation
        • •Find the sequence of rigid transformations
        • •Reflecting figures
        • •Reflecting a point over any non-horizontal line
      • ▸Congruence
        • •Congruent polygons
        • •Congruent supplements and complements theorems
        • •ASA and AAS congruence
        • •Isosceles triangle theorem and its converse
        • •SSS congruence
        • •SAS congruence
        • •Converse of the Pythagorean theorem
        • •Proving isosceles trapezoids
      • ▸Similarity
        • •Intro to similarity
        • •All circles are similar
        • •Finding angles and side lengths in similar polygons
        • •Finding the scale factor for similar polygons
        • •Third angle theorem
        • •Similar triangles
        • •Ratios of area, surface area, and volume, for similar shapes
        • •Crossed ladders theorem
      • ▸Polygons
        • •Perimeter of composite figures
        • •Finding the volume and surface area of composite figures
        • •Sum of the interior angles of a triangle
        • •Sum of interior angles for simple polygons
        • •Sum of exterior angles
        • •Pentagonal tilings
      • ▸Incenter and circumcenter of a triangle
        • •Intro to the incenter and circumcenter of a triangle
        • •Perpendicular bisector theorem and its converse
        • •Circumcenter theorem
        • •Angle bisector theorem
        • •Incenter theorem
      • ▸Triangles
        • •Euler line
        • •Geometric mean theorems
        • •AM-GM inequality
        • •Triangle angle bisector theorem
        • •Ceva's theorem
        • •Proving the altitude of an isosceles triangle cuts two congruent right triangles
        • •Centroid and orthocenter
        • •Ordering triangle sides and angles
        • •Midsegment and proportionality theorems
        • •Menelaus's theorem
        • •Side splitter theorem
        • •Hinge theorem and its converse
        • •Hypotenuse-leg congruence theorem
        • •Hypotenuse-angle congruence theorem
        • •Areas of triangles and quadrilaterals on grids
        • •Exterior angle theorem
      • ▸Quadrilaterals
        • •Diagonals of a kite are perpendicular
        • •Diagonals of a parallelogram bisect each other
        • •Diagonals of a rhombus are perpendicular bisectors
        • •Area of a rhombus from diagonals
        • •Diagonals of a rectangle are congruent
        • •Opposite sides of a parallelogram are congruent
        • •Opposite angles of a parallelogram are congruent
        • •British flag theorem
        • •Cyclic quadrilaterals
        • •Fuhrmann's theorem
        • •Classifying quadrilaterals from four points
      • ▾Circle theorems
        • •Thales's theorem and its converse
        • •Central angle theorem
        • •Angles standing on the same arc are congruent
        • •Chord properties
        • •Congruent tangents theorem
        • •Radius and tangent theorem
        • •Radii inside right triangles
        • •Alternate segments theorem
        • •Power of a point
        • •Three squares puzzle
      • ▸Volume and surface area
        • •Cavalieri's principle
        • •Volume of prisms and cylinders
        • •Volume and surface area of a sphere
      • ▸Analytic geometry
        • •A square within a square
     › High school geometry › Circle theorems

    Three squares puzzle

    Students will learn three ways of solving the three squares geometry puzzle. That is, in the figure below, what is \(\alpha + \beta + \gamma?\)

    Students should be given time to think about how to solve the problem, but most students won't be able to solve it, and after a while, you can give them the hint shown here. After students have received the hint, the puzzle is very doable. After students have found the solution which follows from the hint, show them these additional methods of solution.

    Conclude by giving your students these challenges:

    • Fill Me Up by NRICH
    • Chameleons by NRICH
    • Moving Circles by Pierce School: Problem / Solution

    In this problem, every line is horizontal or vertical, and every line is unique. If there are 2 horizontal lines, and 2 vertical lines, then there are 4 crossings. If there are 3 horizontal lines, and 2 vertical lines, how many crossings are there? If there are at least 2 horizontal lines, and 7 lines in total, what is the least number of crossings? the most? Can you find all possible crossings with 7 lines? Can you find the least/greatest number of crossings for 10 lines? 15? 50? any number of lines?

    Make the equation below true by replacing each letter with a unique digit (0-9).

    $$W + O = OF$$

    Here's the solution:

    The largest sum of two digits is \(9 + 9 = 18,\) so \(O = 1.\) Now we have \(W + 1 = 1F.\) The only way to make \(W + 1 \ge 10\) is if \(W = 9.\) Now we have \(9 + 1 = 1F,\) so \(F = 0.\) In conclusion, the equation is \(9 + 1 = 10.\) Note that it doesn't matter whether we accept leading zeros, such as \(05,\) because if \(O = 0,\) then \(W = F,\) which violates the condition that \(W\) and \(F\) must be unique.

    Lessons and practice problems