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International Contests
Pan-African Shortlist
2017 Pan-African Shortlist
G3
G3
Part of
2017 Pan-African Shortlist
Problems
(1)
Lengths involving regular pentagon and a point on its circumcircle
Source: 2017 Pan-African Shortlist - G3
5/5/2019
Let
A
B
C
D
E
ABCDE
A
BC
D
E
be a regular pentagon, and
F
F
F
some point on the arc
A
B
AB
A
B
of the circumcircle of
A
B
C
D
E
ABCDE
A
BC
D
E
. Show that
F
D
F
E
+
F
C
=
F
B
+
F
A
F
D
=
−
1
+
5
2
,
\frac{FD}{FE + FC} = \frac{FB + FA}{FD} = \frac{-1 + \sqrt{5}}{2},
FE
+
FC
F
D
=
F
D
FB
+
F
A
=
2
−
1
+
5
,
and that
F
D
+
F
B
+
F
A
=
F
E
+
F
C
FD + FB + FA = FE + FC
F
D
+
FB
+
F
A
=
FE
+
FC
.
geometry
pentagon
circumcircle