Problems in Geometry
(4)
1.
Consider a quadrangle
ABCD
with points
P,Q,R,S,T
on
BD,AB,AD,CB,CD
respectively. Prove that if
P,Q,R
are collinear and
P,S,T
are collinear, then the lines
QS,RT,AC
are concurrent.
1
2.
The following is a problem by the late Prof. Demir who liked to present it with a
quaintly arithmetic notation which is very suggestive and delightful. A numeral
m
written
decimally will stand for a point and given such numerals
m,n
the line through them is
denoted by
m
·
n
:
Given distinct non-collinear points 1
,
2
,
3
,
4 consider points 12
∈
1
·
2
- {
1
,
2
}
,
23
∈
2
·
3
- {
2
,
3
}
,
34
∈
3
·
4
- {
3
,
4
}
.
Let 1
·
23 and 12
·
3
,
2
·
34 and 23
·
4 meet in 123
,
234
,
respectively. Finally, let 1
·
234 intersect 123
·
4 in 1234
.
Prove that 12
,
1234
,
34 are
collinear.
2
3.
Given triangle
ABC,
let
D
be a point on
BC,
and
P,Q
points on
AB.
Let
PD
meet
AC
in
H, QD
meet
AC
in
K
and
CP
meet
AD
in
M
and
CQ
meet
AD
in
N.
Prove that
KM
and
HN
meet on
AB.
3