(b)
Let
ABCD
be a cyclic quadrilateral such that the diagonals
AC
and
BD
are perpendicular and
intersect at the point
P
. Let
M
be the midpoint of
AB
and let
N
be the point on
CD
such that
PN
is perpendicular to
CD
. Prove that the points
M
,
P
,
N
lie on a line.
[
4 marks
]
8.
In the hyperbolic plane, there exist pentagons all of whose angles are equal to 90
◦
. Sketch one example
of such a pentagon in the hyperbolic plane using the Poincar
´
e disk model. What is the area of this
pentagon?
[
2 marks
]
9.
Let
D
,
E
,
F
be points on sides
AB
,
BC
,
CA
of triangle
ABC
, respectively. If
DE
=
BE
and
FE
=
CE
,
prove that the circumcentre of triangle
ADF
lies on the angle bisector of
∠
DEF
.
[
BONUS
]
Hints
1.
Draw in the line segments
AI
,
BI
,
CI
or equivalently,
AO
,
BO
,
CO
. What is special about the lengths in the diagram? What is
special about the angles in the diagram?
2.
We already know how to work out all the angles in the diagram with triangle
ABC
and circumcentre
O
. We also already know
how to work out all the angles in the diagram with triangle
ABC
and orthocentre
H
.
3.
(a)
Altitudes lead to right angles and right angles often lead to cyclic quadrilaterals. You should be able to use a cyclic
quadrilateral or two to obtain some useful relations between angles.
(b) Equal chords in a circle subtend equal angles — but remember that equal angles are also subtended by equal chords.
4. Try to mimic the case that we covered in lectures as much as possible.
5.
Since the problem involves a tangent, you should try to use the alternate segment theorem. Use this to prove that
∠
LBX
=
∠
KCX
.
6. How would you prove that triangle
XYB
is isosceles?
7.
(a) Suppose that
C
lies outside the circle with diameter
AB
— can you find a contradiction?
(b) You should use the result from part (a), even if you are unable to prove it.
8. Do you remember how we found the sum of the angles in a polygon with
n
sides?
9. Do you really expect hints for a bonus question?
147

6. ASSIGNMENTS, TESTS AND EXAMINATIONS
Assignment 1 Solutions
1.
Let
ABC
be a triangle with circumcentre
O
and incentre
I
. If
O
and
I
are the same point, prove that the
triangle must be equilateral.
[
1 mark
]
Proof.
Let the angles of triangle
ABC
be 2
a
, 2
b
, 2
c
and recall that
I
lies on the angle bisectors. This
means that
∠
BAI
=
∠
CAI
=
a
,
∠
CBI
=
∠
ABI
=
b
, and
∠
ACI
=
∠
BCI
=
c
.
A
B
C
O
A
B
C
I
a
a
b
b
c
c
If
O
and
I
are the same point, then
AI
=
BI
=
CI
and this means that triangle
ABI
is isosceles. Hence,
∠
BAI
=
∠
ABI
from which it follows that
a
=
b
. We can use the same argument to prove that
b
=
c
, so
all angles of triangle
ABC
are equal. Therefore, triangle
ABC
is equilateral.
2. Let
ABC
be a triangle with circumcentre
O
and orthocentre
H
. Prove that
∠
ABH
=
∠
CBO
.
[
2 marks
]
Proof.
This problem is easy, because we already know that it’s possible to label every angle in the
diagram involving triangle
ABC
and the circumcentre using only
∠
CAB
=
a
,
∠
ABC
=
b
and
∠
BCA
=
c
. We also know that it’s possible to label every angle in the diagram involving triangle
ABC
and the
orthocentre using the same angles
A
B
C
O
A
B
C
H
E
First, note that
BC

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