MathDB
Problems
Contests
International Contests
KoMaL A Problems
KoMaL A Problems 2021/2022
A. 820
A. 820
Part of
KoMaL A Problems 2021/2022
Problems
(1)
Many Concurrencies
Source: KöMaL A. 820
3/23/2022
Let
A
B
C
ABC
A
BC
be an arbitrary triangle. Let the excircle tangent to side
a
a
a
be tangent to lines
A
B
,
B
C
AB,BC
A
B
,
BC
and
C
A
CA
C
A
at points
C
a
,
A
a
,
C_a,A_a,
C
a
,
A
a
,
and
B
a
,
B_a,
B
a
,
respectively. Similarly, let the excircle tangent to side
b
b
b
be tangent to lines
A
B
,
B
C
,
AB,BC,
A
B
,
BC
,
and
C
A
CA
C
A
at points
C
b
,
A
b
,
C_b,A_b,
C
b
,
A
b
,
and
B
b
,
B_b,
B
b
,
respectively. Finally, let the excircle tangent to side
c
c
c
be tangent to lines
A
B
,
B
C
,
AB,BC,
A
B
,
BC
,
and
C
A
CA
C
A
at points
C
c
,
A
c
,
C_c,A_c,
C
c
,
A
c
,
and
B
c
,
B_c,
B
c
,
respectively. Let
A
′
A'
A
′
be the intersection of lines
A
b
C
b
A_bC_b
A
b
C
b
and
A
c
B
c
.
A_cB_c.
A
c
B
c
.
Similarly, let
B
′
B'
B
′
be the intersection of lines
B
a
C
a
B_aC_a
B
a
C
a
and
A
c
B
c
,
A_cB_c,
A
c
B
c
,
and let
C
C
C
be the intersection of lines
B
a
C
a
B_aC_a
B
a
C
a
and
A
b
C
b
.
A_bC_b.
A
b
C
b
.
Finally, let the incircle be tangent to sides
a
,
b
,
a,b,
a
,
b
,
and
c
c
c
at points
T
a
,
T
b
,
T_a,T_b,
T
a
,
T
b
,
and
T
c
,
T_c,
T
c
,
respectively.a) Prove that lines
A
′
A
a
,
B
′
B
b
,
A'A_a,B'B_b,
A
′
A
a
,
B
′
B
b
,
and
C
′
C
c
C'C_c
C
′
C
c
are concurrent.b) Prove that lines
A
′
T
a
,
B
′
T
b
,
A'T_a, B'T_b,
A
′
T
a
,
B
′
T
b
,
and
C
′
T
c
C'T_c
C
′
T
c
are also concurrent, and their point of intersection is on the line defined by the orthocentre and the incentre of triangle
A
B
C
.
ABC.
A
BC
.
Proposed by Viktor Csaplár, Bátorkeszi and Dániel Hegedűs, Gyöngyös
geometry
komal