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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

    Thales's theorem and its converse

    Students will learn about Thales's theorem and its converse. They must already know the isosceles triangle theorem. Our first goal for students, is to have them make the conjecture. Have students begin by opening this file in Geogebra:


    Open with GeoGebra

    Tell them they are allowed to move point \(C\) anywhere along the arc \(AB.\) Ask them what they notice. Alternatively, you could give them a sheet of various triangles in semicircles and a protractor. The conjecture students make may be wrong, but either way, have them attempt to prove it. If they fail to make a conjecture, nudge them along by telling them to concentrate on angle \(ACB.\) After that, challenge your students to prove Thales's theorem. Students are most likely to discover a proof similar to this one. After students have found a proof of Thales's theorem, with or without your help, have them open this Geogebra file:


    Open with Geogebra

    This is nice because it lets students verify the claim for any position of \(C\) they like. It bolsters their confidence that the theorem is true. Doing this isn't strictly necessary, but I think it makes the experience more enjoyable. Next, show your students this proof for the converse of Thales's theorem. Finally, ask students how they could determine the diameter of a circle, given only an additional piece of paper, a pencil, and a ruler. The answer is to place the piece of paper on the circle, such that the corner of the paper lies on the boundary of the circle. Then mark the two points where the edges of the paper cross the circle, using the pencil. Then measure the distance between those points using the ruler (source.)

    Could students discover the proof for the converse of Thales's theorem too, or would that be too difficult for the vast majority of students?

    Conclude by giving your students these challenges:

    • Sum Equals Product by NRICH
    • 2017 Math Kangaroo Level 5-6 Problem #25 by STEM4all
    • 2019 Math Kangaroo Levels 11-12 Problem #16 by STEM4all
    • As Easy as 1,2,3 by NRICH
    • Tangram Money by Pierce School: Problem / Solution

    Lessons and practice problems