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Thales' theorem

    In geometry '''Thales' theorem''' (named after Thales of Miletus) states that if A, B and C are points on a circle where the line AC is a diameter of the circle then the angle ABC is a right angle

    Proof

    We use the following facts: the sum of the angles in a triangleis equal to two right angles and that the base angles ofan isosceles triangle are equal
    Let O be the center of the circle Since OA = OB = OC, OAB and OBC areisosceles triangles and by the equality of the base angles of anisosceles triangle OBC = OCB and BAO = ABO Let γ = BAO and δ = OBC
    Since the sum of the angles of a triangle is equal to two rightangles we have
    2γ + γ ′ = 180°

    and
    2δ + δ ′ = 180°

    We also know that
    γ ′ + δ ′ = 180°

    Adding the first two equations and subtracting the third we obtain
    2γ + γ ′ + 2δ + δ ′ − (γ ′ + δ ′) = 180°

    which after cancelling γ ′ and δ ′ implies that
    γ + δ = 90°

    QED

    Converse

    The converse of Thales' theorem is also valid which states that a right triangle's hypotenuse is a diameter of its circumcircle
    The theorem and its converse can be expressed as follows:
    The center of the circumcircle of a triangle lies on one of the triangle's sides if and only if the triangle is a right triangle

    Proof of the converse

    The proof utilises the fact that directional vectors of two lines form right angles if and only if the dot product is zero Let there be a right angle ABC and circle M with AC as a diameterLet M's center lie on the origin for easier calculationThen the dot product of AB and BC is
    (A − B) · (B − C) = (A − B) · (B + A) = |A|2 − |B|2 = 0

    Hence:
    |A| = |B|

    A and B are also equidistant from the circle's center hence M is the triangle's circumcenter

    Generalization

    Thales' theorem is a special case of the following : given three points A, B and C on a circle with center O, the angle AOC is twice as large as the angle ABC

    History

    Thales was not the first to discover this theorem since the Egyptians and Babylonians must have known of this empirically However they did not prove the theorem and the theorem is named after Thales because he was said to have been the first to prove the theorem using his own results that the base angles of an isosceles triangle are equal and that the sum of angles in a triangle is equal to two right angles

    See also