Bulletin of the Marathwada Mathematicle Society Vol.15, No. 1, June 2014, Pages 1-7
Homothetic Cyclic Quadrilaterals of Cyclic Quadrilateral S. M. Chauthaiwale Department of Mathematics, Amolakchand College, Yavatmal – 445001, India. E-mail:
[email protected]
ABSTRACT This article studies the homothetic nature of the three quadrilaterals corresponding to any cyclic quadrilateral after acknowledging the historical developments in the form of contributions of Indian and Western mathematicians related to the topic.1 1
INTRODUCTION
The properties of cyclic quadrilaterals are usually studied geometrically in terms of its lengths of sides and angles between them without using the concept of coordinates. However this article analytically studies some properties by imposing co-ordinate system on the cyclic quadrilateral. The coordinates of the vertices and of the circumcenter of a given cyclic quadrilateral are first determined in terms of the lengths of its sides. Four triangles formed by diagonals and the sides of this cyclic quadrilateral are considered. Three quadrilaterals are constructed by joining respective centroids, orthocenters and nine-point centers of these triangles. The lengths and slopes of the sides of these quadrilaterals are evaluated after finding the coordinates of each vertex. Finally homothetic inter relations between these quadrilaterals are discussed. We begin with preliminary results stated by Ptolemy, Brahmagupta, Parameshvara, Euler, Poncelet, Brianchon and Feuerbach. These results explain historical developments related to the topic and serve as foreknowledge for said discussion. When not said ABCD is a cyclic quadrilateral. 2
PRELIMINARIES
2.1
Brahmagupta's Results - Let AB = a, BC = b, CD = c and DA = d are lengths of sides
of ABCD then
1
Key words: Cyclic quadrilateral, Homothetic quadrilateral, Euler line, Nine–point circle.
Diagonal AC = √ [
] and
BD = √ [
].
Area of ABCD = √ [(s – a) (s – b) (s – c) (s – d)], where 2 s = a + b + c + d.
(R.1) (R.2)
Note: Brahmagupta is the first mathematician in history of mathematics to state these results. 2.2
Ptolemy's Theorem – For the cyclic quadrilateral ABCD (R.3)
× + × =
× . The converse of this theorem is also true. 2.3 Coordinates of the Vertices of ABCD.
Set the origin at vertex A, X– axis along AB and Y – axis perpendicular to AB. Then coordinates of the vertices are: A (0, 0), B (a, 0), C (AN, CN), D (AM, DM) where CN and DM are perpendiculars from C and D on X-axis. According to a Brahmagupta formula for segments of base by the altitude we have – (Refer Figure 1)
Figure 1
In ∆ ABC, AN = ±
In ∆ ABD, AM = ±
| |
= x1 (say) as per AC > BC or AC < BC and
| | =
x2 (say)
as per BD > AD or BD < AD.
Now CN = √ (AC 2 – AN 2) = y1 (say) and DM = √ (AD 2 – AM 2) = y2 (say). Thus Coordinates of the vertices are: A (0, 0), B (a, 0), C (x1, y1) and D (x2, y2). 2.4
Lengths of Sides and Diagonals in terms of Coordinates - (Refer R.4)
(R.4)
AB = a
BC = √ [(x1 – a) 2 + y12]
CD = √ [(x1 – x2) 2 + (y1 – y2) 2]
DA = √ [x22 + y2 2 ]
AC = √ [x12 + y12]
BD = √ [(x2 – a) 2 + y22]
2.5
Slopes of Sides and Diagonals – (Refer R.4)
Slope of AB = 0, Slope of DA = 2.6
.
Slope of AC =
,
Slope of CD =
Slope BD =
,
Parameshvara's Result: The circumradius ‘R’ of ABCD in terms of its sides is -
R=√ 2.7
Slope of BC =
[ ]
(R.5)
Euler's Result: –The coordinates of the circumcenter of ∆ ABC, with coordinate
system for the triangle set as in Figure 1, areH≡(
,
× !"#
%$[ AC2 + BC2 – AB2 ] ) = ( h1, h2 ) (say).
(R.6)
As points A, B, C and D lie on the same circle, the circumcenter of ∆ BCD, ∆ CDA, ∆ DAB and ABCD is also H ( h1, h2 ). (Refer Figure 1). 2.8
Euler Line –Euler further discovered and proved a very remarkable fact about three
special points related to any triangle that: The circumcenter (H), centroid (G), and orthocenter (E) are collinear and GE = 2 GH, HE = 3 GH.
Figure 2
As this fundamental property and the relations defied Euler’s predecessors for centuries the line HE containing these three points is termed as Euler line for the given triangle Let coordinates be H ≡ (h1, h2), G ≡ (g1, g2), and E ≡ (e1, e2). Then by section formula e1 = 3g1 – 2h1 and e2 = 3g2 – 2h2.
(R.7)
Nine-Point Circle of the Triangle –Poncelet and Brianchon discovered the existence
2.9
of a circle, related to any triangle, passing through nine points (Refer Figure 2) namely three mid-points of sides (M1, M2, M3), three feet of perpendiculars from vertices to opposite sides (M4, M5, M6) and three mid points of segments joining vertices with orthocenter E (M7, M8, M9). Later Feuerbach stated similar results. Hence this circle is also known as Feuerbach circle. The center of this Nine-Point circle not only lies on the Euler line it is midpoint (N) of segment HE and the radius of this circle is half of that of circumradius of the triangle. Thus for nine-point center N ≡ (n1, n2) for the triangle we get n1 =
& '
and n2 =
& '
.
(R.8)
Homothetic Quadrilaterals- Any two quadrilaterals ABCD and PQRS are said to be
2.10
homothetic (Type I) if corresponding sides are proportional and parallel. Thus in Figure 3 – ()
=
)*
=
*+
=
+(
=
, -
and AB || PQ, BC || QR, CD || RS DA || SP.
Figure 3
(R.9)
Figure 4
And any two quadrilaterals ABCD and PQRS are said to be homothetic (Type II) if opposite sides are proportional and parallel. Thus in Figure 4 – *+
=
+(
=
()
=
)*
=
, -
and AB || RS, BC || SP, CD || PQ, DA || QR.
(R.10)
3 Quadrilaterals Joining Centroids, Orthocenters and Nine-point centersThe coordinates of vertices and circumcenter of ABCD are as in R.4 and R.6. Consider ∆ ABC, ∆ BCD, ∆ CDA and ∆ DAB as in Figure 1. Let G1, G2, G3 and G4 respectively denote the centroids, E1, E2, E3, and E4 denote the orthocentres and N1, N2, N3 and N4 denote the nine-point centers of these triangles. Then coordinates of these points are as follows. 3.1
Coordinates of the Centroids are –
G1 ≡ (
G3 ≡ (
3.2
Coordinates of the Orthocenters are –
/
,
/
,
/
),
G2 ≡ (
/
)
G4 ≡ (
,
/ /
,
/
/
),
). (By Centroid formula)
E1 ≡ (a + x1 – 2 h1, y1 – 2 h2),
E2 ≡ (a + x1 + x2 – 2 h1, y1 + y2 – 2 h2)
E3 ≡ (x1 + x2 – 2 h1, y1 + y2 – 2 h2)
E4 ≡ (a + x2 – 2 h1, y2 – 2 h2) (Refer R.7)
3.3
Coordinates of the Nine - point centers are –
N1 ≡
&
N3 ≡
21 + 22 − ℎ1 2
&
,
,
,
N2 ≡
71 + 72 − ℎ2 2
, N4 ≡
1 + 2 1 + 22 − ℎ 1 2 &
,
,
71 + 72 − ℎ2 2
&
,
. (Refer R.8)
These four centroids, orthocenters and nine-point centers are separately joined to construct three quadrilaterals namely G1G2 G3 G4, E1 E2 E3 E4, and N1 N2 N3 N4 (Figure 5). The lengths and the slopes of the sides and of the diagonals of these three quadrilaterals are evaluated and compared with that of ABCD as follows. 3.4
Lengths of the Sides and Lengths of the Diagonals are
G1G2 = / √ [ x2 2 + y2 2 ] =
G3G4 = / √ [(x1 – a)2 + y1 2] = G1 G3 =
/
G2 G3 = / a =
/
/
/
/
G4G1 = √ [(x1 – x2)2 + (y1 – y2)2] =
/
√ [ (x2 – a ) 2 + y2 2 ] =
/
G2G4 = √ [ x12 + y1 2] =
/
. (Refer 3.1 and 2.4)
Similarly E1E2 = DA, E2E3 = AB, E3E4 = BC, E4E1 = CD, E1E3 = BD, E2E4 = AC and N1N2 =
, N2N3 =
, N3N4 =
, N4N1 =
, N1N3 =
, N2N4 =
.
Figure 5 3.5
Slopes of the Sides and Slopes of the Diagonals are -
Slope of G1G2 =
= Slope of DA = Slope of E1E2 = Slope of N1N2.
Slope of G2G3 = 0 = Slope of AB = Slope of E2E3 = Slope of N2N3. Slope of G3G4 = Slope of G4G1 =
= Slope of BC = Slope of E3E4 = Slope of N3N4.
= Slope of CD = Slope of E4E1 = Slope of N4N1.
Slope of G1G3 = Slope of BD = Slope of E1E3 = Slope of N1N3. Slope of G2G4 = Slope of AC = Slope of E2E4 = Slope of N2N4. (Refer 3.1, 3.2, 3.3 and 2.5)
4
Nature of Quadrilaterals G1G2 G3G4, E1E2 E3E4 and N1N2N3N4.
4.1
Comparing the lengths and the slopes of the sides of G1G2 G3G4 and ABCD we get-
8 89 $
=
89 8:
%$=
8: 8 %;
=
8 8 ;
=
/
(Refer 3.4) and
G2G3 || AB, G3G4 || BC, G4G1 || CD and G1G2 || AB. (Refer 3.5) Hence G1G2G3G4 is homothetic (Type II) with ABCD (Refer R.10). Similarly E1E2 E3E4 and N1N2N3N4 are homothetic (Type II) with ABCD as shown in Figure 5. 4.2
