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National and Regional Contests
India Contests
NMTC
2019 NMTC Junior
1
1
Part of
2019 NMTC Junior
Problems
(1)
Midpoints of two sides passing through a point
Source: NMTC 2019 junior P1
11/2/2019
In a convex quadrilateral
P
Q
R
S
PQRS
PQRS
, the areas of triangles
P
Q
S
PQS
PQS
,
Q
R
S
QRS
QRS
and
P
Q
R
PQR
PQR
are in the ratio
3
:
4
:
1
3:4:1
3
:
4
:
1
. A line through
Q
Q
Q
cuts
P
R
PR
PR
at
A
A
A
and
R
S
RS
RS
at
B
B
B
such that
P
A
:
P
R
=
R
B
:
R
S
PA:PR=RB:RS
P
A
:
PR
=
RB
:
RS
. Prove that
A
A
A
is the midpoint of
P
R
PR
PR
and
B
B
B
is the midpoint of
R
S
RS
RS
.
geometry