If D is between A and B, then AD + DB = AB (Segment Addition Postulate). And segment AB has exactly one midpoint which is D (Midpoint Postulate). The midsegment of a triangle is a segment that connects the midpoints of two sides of a triangle. Midsegment Theorem states that the segment that joins the midpoints of two sides of a triangle is parallel to the third side and has a length equal to half the length of the third side. In the figure show above (and below), DE will always be equal to half of BC. Given ? ABC with point D the midpoint of AB and point E the midpoint of AC and point F is the midpoint of BC, the following can be concluded:

Since the tangent of circle is perpendicular to the radius drawn to the tangency point, both radii of the two orthogonal circles A and B drawn to the point of intersection and the line segment connecting the centres form a right triangle. If and are the equations of the two circles A and B, then by Pythagorean theorem, is the condition of the orthogonality of the circles. A Saccheri quadrilateral is a quadrilateral that has one set of opposite sides called the legs that are congruent, the other set of opposite sides called the bases that are disjointly parallel, and, at one of the bases, both angles are right angles.

It is named after Giovanni Gerolamo Saccheri, an Italian Jesuit priest and mathematician, who attempted to prove Euclid’s Fifth Postulate from the other axioms by the use of a reductio ad absurdum argument by assuming the negation of the Fifth Postulate. In hyperbolic geometry, since the angle sum of a triangle is strictly less than radians, then the angle sum of a quadrilateral in hyperbolic geometry is strictly less than radians. Thus, in any Saccheri quadrilateral, the angles that are not right angles must be acute.

Some examples of Saccheri quadrilaterals in various models are shown below. In each example, the Saccheri quadrilateral is labelled as ABCD, and the common perpendicular line to the bases is drawn in blue. For hundreds of years mathematicians tried without success to prove the postulate as a theorem, that is, to deduce it from Euclid’s other four postulates. It was not until the last century or two that four mathematicians, Bolyai, Gauss, Lobachevsky, and Riemann, working independently, discovered that Euclid’s parallel postulate could not be proven from his other postulates.

Their discovery paved the way for the development of other kinds of geometry, called non-Euclidean geometries. Non-Euclidean geometries differ from Euclidean geometry only in their rejection of the parallel postulate but this single alteration at the axiomatic foundation of the geometry has profound effects in its logical consequences. The Lobachevsky geometry is therefore consists of these statements: ? There are lines that are parallel which are everywhere equidistant. ? In any triangle the sum of the three angles is two right angles which is 180 degrees.

? Straight lines parallel to the same line are parallel to each other. ? There exist geometric figures similar with same shape but of different size to other geometric figures. ? Given three points, there is a circle that passes through all three. ? If three angles of a quadrilateral are right angles, then the fourth angle is a right angle. ? There is no triangle in which all three angles are as small as we please. ? There exist squares or equilateral quadrilaterals with four right angles.

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