Geometry

Running Head : GEOMETRY ASSIGNMENT

History of Mathematics - Geometry Assignment

NAME OF CLIENT

NAME OF INSTITUTION

NAME OF PROFESSOR

COURSE NAME

DATE OF SUBMISSION

History of Mathematics - Geometry Assignment (a

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

EF / AB

EF ? AB

DF / AC

DF ? AC

DE / BC

DE ? BC

Therefore , 4 triangles that are congruent are formed (b

Two circles intersecting orthogonally are orthogonal curves and called orthogonal circles of each other

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

is the condition of the orthogonality of the circles (c

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

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

The Beltrami-Klein model

Red lines indicate verification of acute angles by using the poles

The Poincary disc model

The upper half plane model (d

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...