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. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Thales of Miletus According to Bertrand Russell, "Philosophy begins with Thales Thales of Miletus According to Bertrand Russell, "Philosophy begins with Thales Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the Geometry, a diameter of a Circle is any straight Line segment that passes through the center of the circle and whose Endpoints are on the In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called In Geometry and Trigonometry, a right angle is an angle of 90 degrees corresponding to a quarter turn (that is a quarter of a full circle
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Proof
We use the following facts:
- the sum of the angles in a triangle is equal to two right angles (180°),
- the base angles of an isosceles triangle are equal. A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line This article describes the unit of angle For other meanings see Degree. A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line
Let O be the center of the circle. Since OA = OB = OC, OAB and OBC are isosceles triangles, and by the equality of the base angles of an isosceles triangle, OBC = OCB and BAO = ABO. Let α = BAO and β = OBC. The 3 internal angles of the ABC triangle are α, α + β and β. Since the sum of the angles of a triangle is equal to two right angles, we have
then
or simply
Converse
The converse of Thales' theorem is also valid; it states that a right triangle's hypotenuse is a diameter of its circumcircle. QED is an abbreviation of the Latin phrase "la '''quod erat demonstrandum'''" which means literally "that which was to be demonstrated" Two types of special right triangles appear commonly in geometry the "angle based" and the "side based" (or Pythagorean Triangles The former are characterised A hypotenuse is the longest side of a Right triangle, the side opposite of the Right angle. In Geometry, the circumscribed circle or circumcircle of a Polygon is a Circle which passes through all the vertices of the polygon
Combining Thales' theorem with its converse we get that:
- The center of a triangle's circumcircle lies on one of the triangle's sides if and only if the triangle is a right triangle. ↔
Proof of the converse using geometry
This proof consists of 'completing' the right triangle to form a rectangle and noticing that the center of that rectangle is equidistant from the vertexes and so is the center of the circumscribing circle of the original triangle, it utilizes two facts:
- adjacent angles in a parallelogram are supplementary (add to 180°) and,
- the diagonals of a rectangle are equal and cross each other in their medium point. In Geometry, a rectangle is defined as a Quadrilateral where all four of its angles are Right angles A rectangle with vertices ABCD would be denoted as In Geometry, a parallelogram is a Quadrilateral with two sets of Parallel sides This article describes the unit of angle For other meanings see Degree.
Let there be a right angle ABC, r a line parallel to BC passing by A and s a line parallel to AB passing by C. Let D be the point of intersection of lines r and s.
The quadrilateral ABCD forms a parallelogram by construction (as opposite sides are parallel). Since in a parallelogram adjacent angles are supplementary (add to 180°) and ABC is a right angle (90°) then angles BAD, BCD, and ADC are also right (90°); consequently ABCD is a rectangle.
Let O be the point of intersection of the diagonals AC and BD. Then the point O, by the second fact above, is equidistant from A,B,C, and D. And so O is center of the circumscribing circle, and the hypotenuse of the triangle AC is a diameter of the circle.
Proof of the converse using linear algebra
This proof utilizes two facts:
- two lines form a right angle if and only if the dot product of their directional vectors is zero, and
- the square of the length of a vector is given by the dot product of the vector with itself. In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R
Let there be a right angle ABC and circle M with AC as a diameter. Let M's center lie on the origin, for easier calculation. Then we know
- A = − C, because the circle centered at the origin has AC as diameter, and
- (A − B) · (B − C) = 0, because ABC is a right angle.
It follows
- 0 = (A − B) · (B − C) = (A − B) · (B + A) = |A|2 − |B|2.
Hence:
- |A| = |B|.
This means that A and B are equidistant from the origin, i. e. from the center of M. Since A lies on M, so does B, and the circle M is therefore the triangle's circumcircle.
The above calculations in fact establish that both directions of Thales' theorem are valid in any inner product space. In Mathematics, an inner product space is a Vector space with the additional Structure of inner product.
Generalization
Thales' theorem is a special case of the following theorem:
- Given three points A, B and C on a circle with center O, the angle AOC is twice as large as the angle ABC.
See inscribed angle, the proof of this theorem is quite similar to the proof of Thales' theorem given above. In Geometry, an inscribed angle is formed when two Secant lines of a Circle (or in a Degenerate case, when one Secant line and
Application
Thales' theorem can be used to construct the tangent to a given circle that passes through a given point. For the tangent function see Trigonometric functions. For other uses see Tangent (disambiguation. (See figure. ) Given a circle k, with a center O, and a point P outside of the circle, we want to construct the (red) tangent(s) to k that pass through P. Suppose the (as yet unknown) tangent t touches the circle in the point T. From symmetry, it is clear that the radius OT is orthogonal to the tangent. So construct the midpoint H between O and P, and draw a circle centered at H through O and P. By Thales' theorem, the sought point T is the intersection of this circle with the given circle k, because that is the point on k that completes a right triangle OTP.
Since there the two circle intersect in two points, we can construct both tangents in this fashion.
History
Thales was not the first to discover this theorem since the Egyptians and Babylonians must have known of this empirically, however there is no record of a proof of the theorem by either of them. 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. A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line
See also
External links
- Munching on Inscribed Angles
- Thales' theorem explained With interactive animation
- Thales' Theorem by Michael Schreiber, The Wolfram Demonstrations Project. Synthetic geometry is the branch of Geometry which makes use of Theorems and synthetic observations to draw conclusions as opposed to Analytic geometry
- Eric W. Weisstein, Thales' Theorem at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
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