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Nepal National Olympiad
2018 Nepal National Olympiad
3a
3a
Part of
2018 Nepal National Olympiad
Problems
(1)
Nepal National Olympiad
Source: Nepal Mathematics Olympiad
4/22/2018
Problem Section #3 a) Circles
O
1
O_1
O
1
and
O
2
O_2
O
2
interest at two points
B
B
B
and
C
C
C
, and
B
C
BC
BC
is the diameter of circle
O
1
O_1
O
1
. Construct a tangent line of circle
O
1
O_1
O
1
at
C
C
C
and intersecting circle
O
2
O_2
O
2
at another point
A
A
A
. Join
A
B
AB
A
B
to intersect circle
O
1
O_1
O
1
at point
E
E
E
, then join
C
E
CE
CE
and extend it to intersect circle
O
2
O_2
O
2
at point
F
F
F
. Assume
H
H
H
is an arbitrary point on line segment
A
F
AF
A
F
. Join
H
E
HE
H
E
and extend it to intersect circle
O
1
O_1
O
1
at point
G
G
G
, and then join
B
G
BG
BG
and extend it to intersect the extend of
A
C
AC
A
C
at point
D
D
D
. Prove:
A
H
H
F
=
A
C
C
D
\frac{AH}{HF}=\frac{AC}{CD}
H
F
A
H
=
C
D
A
C
.
geometry