MathDB
Problems
Contests
International Contests
IMO Shortlist
2018 IMO Shortlist
G5
G5
Part of
2018 IMO Shortlist
Problems
(1)
Triangle form by perpendicular bisector
Source: IMO Shortlist 2018 G5
7/17/2019
Let
A
B
C
ABC
A
BC
be a triangle with circumcircle
Ω
\Omega
Ω
and incentre
I
I
I
. A line
ℓ
\ell
ℓ
intersects the lines
A
I
AI
A
I
,
B
I
BI
B
I
, and
C
I
CI
C
I
at points
D
D
D
,
E
E
E
, and
F
F
F
, respectively, distinct from the points
A
A
A
,
B
B
B
,
C
C
C
, and
I
I
I
. The perpendicular bisectors
x
x
x
,
y
y
y
, and
z
z
z
of the segments
A
D
AD
A
D
,
B
E
BE
BE
, and
C
F
CF
CF
, respectively determine a triangle
Θ
\Theta
Θ
. Show that the circumcircle of the triangle
Θ
\Theta
Θ
is tangent to
Ω
\Omega
Ω
.
geometry