Download Quadrilaterals

Quadrilaterals
MA 341 – Topics in Geometry
Lecture 22
Theorems
1. A convex quadrilateral is cyclic if and
only if the four perpendicular bisectors
of the sides are concurrent.
2 A convex quadrilateral
2.
d il t
l iis cyclic
li if and
d
only if opposite angles are
supplementary.
l
t
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Ptolemy’ss Theorem
Ptolemy
Let a, b, c, and d be the
l
lengths
h of
f consecutive
sides of a cyclic
quadrilateral and let x
and y be the lengths of
the diagonals. Then
ac + bd = xy.
Also,
b
y
a
x
c
d
x ad + bc
=
y ab + cd
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Ptolemy’ss Theorem
Ptolemy
This leads to the
f ll
following:
b
((ac +bd)(ad
)( + bc))
x=
ab + cd
y
a
x
c
and
(ac + bd)(ab + cd)
y=
ad
d +b
bc
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4
Angles of a cyclic quadrilateral
If A is the angle between
a and
dd
a2 + d2 -b2 - c2
cos(A) =
2(ad + bc)
( - a)(s
(s
)( -b)(s
b)( - c)(s
)( - d)
sin(A) = 2
ad +bc
A
(s - a)(s - d)
tan =
2
(s -b)(s
b)(s - c)
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Angles of a cyclic quadrilateral
The angle, θ, between the diagonals
(s -b)(s - d)
tan =
2
(s - a)(s - c)

The circumradius is
1 (ab + cd)(ac + bd)(ad + bc)
R=
4
(s - a)(s -b)(s - c)(s - d)
(P
(Parameshvara’s
h
’ formula)
f
l )
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Incenters of the triangles
The incenters of 4
triangles form a
rectangle
Sides parallel
Sid
ll l tto
lines through
midarc
id
points
i t
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Incenters of the triangles
The incenters and
excenters of 4
triangles form a
4 x 4 grid
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Incenters of the triangles
Consider
quadrilaterals
formed by
centroids, 9-point
circle centers and
orthocenters of
the 4 triangles
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Incenters of the triangles
All are similar to
ABCD.
H2H1H4H3 is
congruentt to
t ABCD
ABCD.
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Tangential Quadrilaterals
A convex
quadrilateral whose
sides all lie tangent
to inscribed circle
– the incircle
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Tangential Quadrilaterals
Is a rectangle a tangential quadrilateral?
Is a square a tangential quadrilateral?
Is a parallelogram
ll l
a tangentiall
quadrilateral?
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Tangential Quadrilaterals
Is a rhombus a tangential quadrilateral?
Is a kite a tangential quadrilateral?
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Tangential Quadrilaterals
In triangle incenter
= intersection of
angle bisectors
Whatt about
Wh
b t
quadrilaterals?
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Tangential Quadrilaterals
I is equidistant from AD,
AB and BC
B
∆API  ∆AMI  PI  MI
N
C
M
∆MIB  ∆NIB  NI  MI
D
P
A
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Tangential Quadrilaterals
I is equidistant from AD,
AB BC and CD
AB,
∆API  ∆AMI  PI  MI
B
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C
M
∆MIB  ∆NIB  NI  MI
etc
Thus, there is an inscribed
circle!
A
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N
D
P
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Tangential Quadrilaterals
If there is an
inscribed circle,
then center is
equidistant from
sides.
sides
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Tangential Quadrilaterals
Then, BI,
Then
BI CI,
CI DI,
DI
and AI are angle
bisectors.
bisectors
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Theorem
A quadrilateral is
tangential if and
only if
a + c = b + d.
N
M
O
P
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Proof
BM=BN
CN=CO
DO DP
DO=DP
M
AP=AM
a + c = AM+MB+CO+OD
= AP+BN+CN+DP
=b+d
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N
O
P
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Proof
Assume a + c = b + d.
B
B’
Construct
C
st
t circle
i l tangent
t
t
to AB, AD, CD.
b
Construct CB’’ tangent to
circle
WLOG AB’<AB
a
A
d
D
c
C
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Proof
AB’CD
D tangential
g

AB’ + c = d + B’C
a +c=d+b
Subtraction
a - AB
AB’ = b – B
B’C
C
BB’ + B’C = b
B=B
B’
 ABCD tangential
B
B’
a
A
d
b
D
c
C
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Result
If ABCD tangential quadrilateral then
a +b + c + d
a +c =b +d =
=s
2
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Area
For inradius r and
semiperimeter s
N
K = rs
M
O
P
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Area
Trigonometric formula
p
K = rs
D
 B +D
K = abcd
b d sin
i 

 2 
q
1 2 2
2
K=
p q - (ac -bd)
2
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Further Results
1 If a line cuts a tangential quadrilateral
1.
into two polygons with equal areas and
equal perimeters,
perimeters the line passes
through the incenter
2 If M and N are midpoints of diagonals
2.
and incenter = I, then M, I, and N are
collinear
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Further Results
3 If incircle is tangent to AB,
3.
AB BC,
BC CD,
CD DA
at X, Y, Z, W respectively, then lines
XY WZ and AC are concurrent
XY,
4. If I is incenter then
IAIC + IBID = ABBC CDDA
5. Incenter I of ABCD coincides with
centroid of ABCD if and only if
IAIC = IBID
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Further Results
Look at the incircles of the triangles of a
tangential quadrilateral.
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Further Results
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