MathDB
Problems
Contests
National and Regional Contests
Russia Contests
Sharygin Geometry Olympiad
2022 Sharygin Geometry Olympiad
17
17
Part of
2022 Sharygin Geometry Olympiad
Problems
(1)
All lines PQ concur
Source: Sharygin 2022 P17
3/4/2022
Let a point
P
P
P
lie inside a triangle
A
B
C
ABC
A
BC
. The rays starting at
P
P
P
and crossing the sides
B
C
BC
BC
,
A
C
AC
A
C
,
A
B
AB
A
B
under the right angle meet the circumcircle of
A
B
C
ABC
A
BC
at
A
1
A_{1}
A
1
,
B
1
B_{1}
B
1
,
C
1
C_{1}
C
1
respectively. It is known that lines
A
A
1
AA_{1}
A
A
1
,
B
B
1
BB_{1}
B
B
1
,
C
C
1
CC_{1}
C
C
1
concur at point
Q
Q
Q
. Prove that all such lines
P
Q
PQ
PQ
concur.
geometry