Modern Geometry Topics

MODERN GEOMETRY
Group 4 Presentation
 Triangles and Quadrangles
 Definitions
 Desarguesian Plane
 Harmonic Sequence of
Points/Lines

TRIANGLES
 Definition
 Types
 Properties
 Secondary part
 Congruency
 Area

 A 3-sided polygon that has 3 sides, 3 vertices
and 3 angles. There are 3 types of triangle base
on its sides, equilateral, isosceles and scalene.
 All triangles are convex and bicentric. That
portion of the plane enclosed by the triangle is
called the triangle interior, while the remainder
is the exterior.
 The study of triangles is sometimes known as
triangle geometry and is a rich area of
geometry filled with beautiful results and
unexpected connections.
Basic Definition of Triangles

On Length of Sides
1) Equilateral
Triangle
2) Isosceles Triangle
3) Scalene Triangle
Types of Triangles
On Basis of Angles
1) Acute Angled Triangle
2) Obtuse Angled
Triangle
3) Right Angled Triangle

EQUILATERAL TRIANGLE
Triangles having all sides equal
Basis of Length of Sides

ISOSCELES TRIANGLE
Triangles having 2 sides equal

SCALENE TRIANGLE
Triangles having no sides equal

Basis of Angles
ACUTE ANGLED TRIANGLE
Triangles whose all angles are acute
angle.
Note: Acute angles are angles less than 90.

Basis of Angles
OBTUSE ANGLED TRIANGLE
Triangles whose one (1) is obtuse
angle.
Note: Obtuse angles are angles more than 90.

Basis of Angles
RIGHT ANGLED TRIANGLE
Triangles whose one (1) angle is right
angle.
Note: Right angles are angles equal to 90. All Right Angles are equal (4th
Postulate)

Properties of a Triangle
 Triangles are assumed to be 2D plane
figures, unless the context provides
otherwise.
 In rigorous treatments, a triangle is
therefore called a 2-simplex. Elementary
facts about triangles were presented by
Euclid in books 1–4 of his Elements,
around 300 BC.
 The interior angles of the triangle always
add up to 180 degrees.

The measures of the interior angles of a
triangle in Euclidean space always add up
to 180 degrees.
mA+ mB+ mC = 180
Exterior angle of a triangle is an angle
that is a linear pair to an interior angle.

Exterior Angle Theorem
States that the measure of an exterior
angle of a triangle is equal to the sum of
the measures of the two interior angles
that are not adjacent to it.
The sum of the measures of the three
exterior angles (one for each vertex) of
any triangle is 360 degrees.

Angle Sum Property
States that the sum of all
interior angles of a triangle
is equal to 180˚.
Exterior Angle Property
States that an exterior
angle of the triangle is
equal to sum of two
opposite interior angles of
the triangle. mA+ mB= mC
mA+ mB+ mC=180

PYTHAGOREAN
THEOREM
States that in a right
angled triangle, the
square of the
hypotenuse is equal
to the sum of squares
of the rest of the two
sides.
Note: This theorem is named after Pythagoras of Samos a Greek
mathematician and philosopher (570 BCE).
(hypotenuse)2= (side)2+(side)2
(c)2= (a)2+(b)2

 Median
 Altitude
 Perpendicular Bisector
 Angle Bisector
Secondary Parts of A Triangle

 Median
Is the line segment
joining the midpoint
of the base of the
triangle.
MEDIAN
MIDPOINT
BAS
E

 Altitude
The line segment
drawn from a vertex
of a triangle
perpendicular to its
opposite.
ALTITUDE
BAS
E

 Perpendicular
Bisector
A line that passes
through midpoint of
the triangle or the
line which bisects
the third side of the
triangle and is
perpendicular to it
Note: Perpendicular is the
relationship between
two lines which meet at a
right angle.
Perpendicular
Bisector/s

 Angle Bisector
A line segment that
bisects an angle of a
triangle.
Note: Bisect means to divide into two equal
parts
Angle Bisector/s

 Side-Side-Side (SSS)
 Side-Angle-Side (SAS)
 Angle-Side-Angle
(ASA)
 Right Angle-
Hypotenuse-Side
(RHS)
Congruence of A Triangle

In above given figure, AB= PQ,
QR= BC and AC=PR, hence D
ABC ≅ D PQR.
SSS (Side-Side-Side)
If all the three sides of
one triangle are
equivalent to the
corresponding three
sides of the second
triangle, then the two
triangles are said to be
congruent by SSS rule.

In above given figure, sides AB=
PQ, AC=PR and angle between
AC & AB equal to angle between
PR and PQ i.e. Ð A = Ð P, hence D
ABC ≅ D PQR.
SAS (Side-Angle-
Side):
If any 2 sides and angle
included between the sides
of one triangle are
equivalent to the
corresponding two sides
and the angle between the
sides of the second triangle,
then the 2 triangles are said
to be congruent by SAS
rule.

In above given figure, Ð B = Ð
Q,
Ð C = Ð R and sides between Ð
B and ÐC , Ð Q and Ð R are
equal to each other i.e. BC= QR,
hence D ABC ≅ D PQR.
ASA (Angle-Side- Angle)
If any two angles and side
included between the
angles of one triangle are
equivalent to the
corresponding two angles
and side included between
the angles of the second
triangle, then the two
triangles are said to be
congruent by ASA rule.

In above figure, hypotenuse XZ =
RT and side YZ=ST, hence
triangle XYZ ≅ triangle RST.
RHS (Right angle-
Hypotenuse-Side)
If the hypotenuse and a
side of a right- angled
triangle is equivalent to the
hypotenuse and a side of
the second right- angled
triangle, the right triangles
are said to be congruent by
RHS rule.

Area of A Triangle
Heron’s Formula can be used in finding area of
all types of Triangles.
Step 1: Calculate "s"
Step 2: Then calculate the Area
Area for Isosceles
Triangle
and Right Triangle
½ (base)x(height) or ½ bh

QUADRANGLES
 Basic Definition of Quadrangles
 Complete Quadrangle
 Diagonals
 Diagonal Triangle
 Projective Properties
 Euclidean Properties

QUADRANGLES
 A quadrangle is a polygon that has 4
sides. Another name for quadrangle is
quadrilateral. The prefix “quad-”
means four. All quadrangles have 4
sides, 4 vertices, and 4 angles.

Properties of A Quadrangle
 All quadrangles are 2D.
 All quadrangles have 4
vertices.
 All quadrangles have 4 sides.
 All quadrangles have 4
angles.
Basic Definition of Quadrangles

NOTE:
Unlike triangles, quadrangles are
not rigid. Their shapes and areas
may easily be changed. But for
any given quadrangle, the
perimeter does not change and, for
any given parallelogram, opposite
sides remain parallel in spite of the
changes to their inner angles.

COMPLETE QUADRANGLE
 A complete quadrangle is a
set of four points, no three of
which are collinear, and the six
lines incident with each pair of
these points. The four points
are called vertices and the six
lines are called sides of the
quadrangle.

 A complete quadrilateral is a system of four lines,
no three of which pass through the same point,
and the six points of intersection of these lines.
 The complete quadrangle (a) was called
a TETRASTIGM by Lachlan (1893), and the
complete quadrilateral (b) was called
a TETRAGRAM.
COMPLETE QUADRANGLE
(a) (b)

Diagonals
 The six lines of a complete
quadrangle meet in pairs to form
three additional points called
the diagonal points of the
quadrangle. Among the six points of a
complete quadrilateral there
are three pairs of points that
are not already connected by
lines, the line
segments connecting these
pairs are called diagonals.

Diagonal Triangle
 The diagonal triangle of
a complete quadrangle is the
triangle formed by its three
diagonal points.
 If the quadrangle is a cyclic
quadrilateral, then the circle is
the polar circle of the diagonal
triangle, i.e., each vertex is
the inversion pole of the
opposite side with respect to
the circle.

Projective properties
 The complete quadrangle and
the complete quadrilateral both
form projective configurations.
 The notation of projective
configurations in TETRASTIGM
is written as (4362) and
TETRAGRAM is written
(6243),where the numbers in this
notation refer to the numbers of
points, lines per point, lines, and
points per line of the

 TETRASTIGM is
(4362)
 TETRAGRAM is
(6243)
Points
Lines per point
Lines
Points per line

 The projective dual of a complete
quadrangle is a complete quadrilateral,
and vice versa.
 For any two complete quadrangles, or
any two complete quadrilaterals, there is
a unique projective transformation
taking one of the two configurations into
the other.

 When each pair of opposite
sides of the quadrangle intersect
on a line, then the diagonals
intersect the line at projective
harmonic conjugate positions.
 The four points on the line
deriving from the sides and
diagonals of the quadrangle are
called a harmonic range.
Through perspectivity and
projectivity, the harmonic
property is stable. - Karl
von Staudt
KLMN is complete
quadrangle
D is projective harmonic
conjugate of C.

Euclidean properties
 Wells (1991) describes several additional
properties of complete quadrilaterals that
involve metric properties of the Euclidean
plane, rather than being purely projective.
 The midpoints of the diagonals are
collinear, and also collinear with the center
of a conic that is tangent to all four lines of
the quadrilateral.

 Any three of the lines of the quadrilateral form
the sides of a triangle.
 The Orthocenters of the four triangles formed in
this way lie on a second line, perpendicular to
the one through the midpoints.

 The Circumcircles of these
same four triangles meet in a
point. In addition, the three
circles having the diagonals as
diameters belong to a
common pencil of circles the
axis of which is the line through
the orthocenters.
 The Polar Circles of the
triangles of a complete
quadrilateral form
a coaxal system.

Desarguesian Plane
 Desarguesian Plane Definition
 Non-Desarguesian Plane
 Desarguesian Plane
 Desarguesian Projective Plane
 Additional Notes

Desarguesian Plane Definition
Desarguesian Plane (named after Girard Desargues)
 Is a plane that can be constructed from a
three-dimensional vector space over a skew
field.
 A projective plane is Desarguesian if
Desargues' theorem holds for this plane.
 The PG(2,K) notation is reserved for the
Desarguesian planes.

Non- Desarguesian plane
 A projective plane that does not
satisfy Desargues' theorem, or in other words a
plane that is not a Desarguesian plane.
Some of the known examples of non-
Desarguesian planes include:
 The Moulton plane
 Hughes planes
 Hall planes
 André planes
Non-Desarguesian Plane

Desarguesian Plane
 Every Desarguesian plane can be
constructed from a skew-field F, by working
in the vector space F³, discarding the origin
and identifying scalar multiples of a point.
The converse is also true.
 If a plane can be embedded in a three-
dimensional projective space, it is
automatically Desarguesian.

Desarguesian Projective Plane
 A finite projective plane is Desarguesian if it
has a group of collineations that acts doubly-
transitively on its points. The group of
collineations of a Desarguesian projective
plane PG (2, ph) has order
 The group of collineations of a non-
Desarguesian projective plane PG (2, n) has
order at most

Additional Notes
 A projective plane is called Desarguesian if
the Desargues assumption holds in it (i.e. if it
is isomorphic to a projective plane over a
skew-field).
 The idea of finite projective planes (and
spaces) was introduced by K. von Staudt.
 The fact that a finite projective plane with
doubly-transitively acting group of
collineations is Desarguesian is the Ostrom–
Wagner theorem.

Additional Notes
 A finite Desarguesian is Pappian, since a finite
skew-field is commutative.
 A collineation of order of is called a Singer
cycle, It has now been settled by exhaustive
computer search that there are no projective
planes of order 10, and precisely four non-
isomorphic projective planes of order 9.
 The algebraic structure coordinatizing a
projective plane is usually called a planar
ternary ring. The flag-transitive finite projective
planes have been determined by W.M. Kantor

Harmonic Sequence
 Basic Definition of Harmonic
Sequence
 Harmonic Sequence Use in
Geometry
 Harmonic Sequence of
Points/Lines

Basic Definition of Harmonic Sequence
Harmonic Sequence (or Harmonic Progression)
A progression formed by taking the
reciprocals of an arithmetic progression. In
other words, it is a sequence of the form
where −a/d is not a natural
number and k is a natural number.

Basic Definition of Harmonic Sequence
 A sequence is a harmonic progression
when each term is the harmonic mean of
the neighboring terms.
 It is not possible for a harmonic
progression to sum to an integer.

Harmonic Sequence Use in Geometry
 If collinear points A, B, C, and D are such that D
is the harmonic conjugate of C with respect to A
and B, then the distances from any one of these
points to the three remaining points form
harmonic progression.
 Each of the sequences AC, AB, AD; BC, BA, BD;
CA, CD, CB; and DA, DC, DB are harmonic
progressions, where each of the distances is
signed according to a fixed orientation of the line.
 In a triangle, if the altitudes are in arithmetic
progression, then the sides are in harmonic
progression.

 The concept of a complete quadrangle is
used to define the concept of a
harmonic sequence of points on a line.
 Four points A, B, C, and D on a line are
said to form a harmonic sequence if A
and B are diagonal points of a complete
quadrangle, and C and D are on the
sides of the quadrangle that meet at the
third diagonal point.
Harmonic Sequence of Points/Lines

 By taking the dual of this definition we
obtain the definition of the dual concept
of a harmonic sequence of line on a
point.
 If A, B, C and D form a harmonic
sequence, we say that C and D are
harmonic conjugates with respect to A
and B.

 It is clear from the definition of a
harmonious sequence that the first and
second members of the sequence may
be interchanged and the third and fourth
members may be interchanged.
 That is, if A, B, C, D is a harmonic
sequence, so are B, A, C, D and A, B, D,
C.

 It can be shown, too, that the first pair
and the second pair of the harmonic
sequence may be interchange.
 That is, if A, B, C, D is a harmonic
sequence, then so is C, D, A, B.

 If A, B and C are any three distinct
points on the line, there is a point D on
the line such that the four points A, B, C
and D form a harmonic sequence.

Theorem (1)
A projection of a harmonic sequence of
points on a line is a harmonic sequence of
lines on a point.
The converse of this theorem is its dual, so it is also a valid theorem.
Theorem (2)
A section of a harmonic sequence of lines
on a point is a harmonic sequence of
points on a line.

Conclusion:
By using the two theorems we can
establish that if two sequence of a points A,
B, C, D and A’, B’, C’, D’ are perspective
from a point, and one of the sequences is
harmonic, then so is the other.

Proof:
 Let O be the center of perspectivity, and
denote OA, OB, OC and OD by a, b, c and d
respectively.
 If A, B, C, and D form a harmonic sequence,
then so do their projections, a, b, c and d.
 But then, since a, b, c and d form a harmonic
sequence, so do their sections A’, B’, C’, and
D’.

References:
 https://en.wikipedia.org/wiki/Complet
e_quadrangle
 https://en.wikipedia.org/wiki/Altitude_
(triangle)#Orthocenter
 http://www.mathwarehouse.com/geo
metry/triangles/area/index.php
 https://cp4space.wordpress.com/201
3/09/01/pappian-and-desarguesian-
planes/

HYPERLINKS - Notes
Proceed to Properties
Slide 4 with hyperlinks (Types of Triangles)
Return to Slide 4
Proceed to Area of Triangle
Return to Slide 21
Slide 16 with hyperlinks (Secondary Parts of A Triangle)
Proceed to Congruence
Return to Slide 16
Slide 21 with hyperlinks (Congruence of A
Triangle)
Return to Slide 45
Note: You can remove/use
the background sound in the
First Slide of the
Presentation 

Prepared by:
Group 4
Members:
Llagas, Elma M.
Anay, Anthony
Ciano, Joshua C.
Beleno, Norma S.
Bobis, Gerald
Beriña, Paula Jean
Rivera, Julie V.

DESARGUES’ THEOREM
 Two triangles are in perspective about a point if
and only if they are in perspective about a line.
 Equivalently, suppose we have a collection of 10
points and 10 lines, associating an ordered
pair (p(V), l(V)) of a point and a line to
each vertex V of the Petersen graph, where each
edge is replaced with a zweibeck (pair of
oppositely directed edges). We consider
each directed edge X → Y to represent the
point p(X) and the line l(Y). Desargues’ theorem
states that if 29 of these correspond to
incidences (a point lying on a line), then so does
the thirtieth

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