consultantnero.blogg.se - Parallelogram abcd

Given a right triangle ABC with hypotenuse AC, construct a circle Ω whose diameter is AC. And so O is center of the circumscribing circle, and the hypotenuse of the triangle ( AC) is a diameter of the circle.Īlternate proof of the converse using geometry Then the point O, by the second fact above, is equidistant from A, B, and C. Let O be the point of intersection of the diagonals AC and BD.

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. The quadrilateral ABCD forms a parallelogram by construction (as opposite sides are parallel). Let D be the point of intersection of lines r and s (Note that it has not been proven that D lies on the circle) 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.

the diagonals of a rectangle are equal and cross each other in their median point.

adjacent angles in a parallelogram are supplementary (add to 180 °) and,.

This proof consists of 'completing' the right triangle to form a rectangle and noticing that the center of that rectangle is equidistant from the vertices and so is the center of the circumscribing circle of the original triangle, it utilizes two facts: (Equivalently, a right triangle's hypotenuse is a diameter of its circumcircle.) The converse of Thales's theorem is then: the center of the circumcircle of a right triangle lies on its hypotenuse. One way of formulating Thales's theorem is: if the center of a triangle's circumcircle lies on the triangle then the triangle is right, and the center of its circumcircle lies on its hypotenuse. This point must be equidistant from the vertices of the triangle.) This circle is called the circumcircle of the triangle. The perpendicular bisectors of any two sides of a triangle intersect in exactly one point. The locus of points equidistant from two given points is a straight line that is called the perpendicular bisector of the line segment connecting the points. All angles in a rectangle are right angles.įor any triangle, and, in particular, any right triangle, there is exactly one circle containing all three vertices of the triangle. Α + ( α + β ) + β = 180 ∘, the diagonals of the parallelogram, are both diameters of the circle and therefore have equal length, the parallelogram must be a rectangle. Since the sum of the angles of a triangle is equal to 180°, we have

The three internal angles of the ∆ABC triangle are α, ( α + β), and β. Since OA = OB = OC, ∆OBA and ∆OBC are isosceles triangles, and by the equality of the base angles of an isosceles triangle, ∠OBC = ∠OCB and ∠OBA = ∠OAB.

The following facts are used: the sum of the angles in a triangle is equal to 180 ° and the base angles of an isosceles triangle are equal. The theorem is named after Thales because he was said by ancient sources 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 of a triangle is equal to a straight angle (180°).ĭante's Paradiso (canto 13, lines 101–102) refers to Thales's theorem in the course of a speech. It is believed that Thales learned that an angle inscribed in a semicircle is a right angle during his travels to Babylon. Reference to Thales was made by Proclus, and by Diogenes Laërtius documenting Pamphila's statement that Thales "was the first to inscribe in a circle a right-angle triangle".īabylonian mathematicians knew this for special cases before Thales proved it. Attribution did tend to occur at a later time.

Work done in ancient Greece tended to be attributed to men of wisdom without respect to all the individuals involved in any particular intellectual constructions this is true of Pythagoras especially. There is nothing extant of the writing of Thales. – English translation by Henry Wadsworth Longfellow – Dante's Paradiso, Canto 13, lines 100–102