MathDB
Problems
Contests
International Contests
Pan-African Shortlist
2018 Pan-African Shortlist
G1
G1
Part of
2018 Pan-African Shortlist
Problems
(1)
2018 PAMO Shortlist: Circle with diameter AC, angle bisector, midsegment concur
Source: 2018 Pan-African Shortlist - G1
5/6/2019
In a triangle
A
B
C
ABC
A
BC
, let
D
D
D
and
E
E
E
be the midpoints of
A
B
AB
A
B
and
A
C
AC
A
C
, respectively, and let
F
F
F
be the foot of the altitude through
A
A
A
. Show that the line
D
E
DE
D
E
, the angle bisector of
∠
A
C
B
\angle ACB
∠
A
CB
and the circumcircle of
A
C
F
ACF
A
CF
pass through a common point.Alternate version: In a triangle
A
B
C
ABC
A
BC
, let
D
D
D
and
E
E
E
be the midpoints of
A
B
AB
A
B
and
A
C
AC
A
C
, respectively. The line
D
E
DE
D
E
and the angle bisector of
∠
A
C
B
\angle ACB
∠
A
CB
meet at
G
G
G
. Show that
∠
A
G
C
\angle AGC
∠
A
GC
is a right angle.
geometry
angle bisector