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National and Regional Contests
Bulgaria Contests
Bulgaria National Olympiad
1991 Bulgaria National Olympiad
Problem 1
Problem 1
Part of
1991 Bulgaria National Olympiad
Problems
(1)
point on triangle altitude, projection onto sides
Source: Bulgaria 1991 P1
6/2/2021
Let
M
M
M
be a point on the altitude
C
D
CD
C
D
of an acute-angled triangle
A
B
C
ABC
A
BC
, and
K
K
K
and
L
L
L
the orthogonal projections of
M
M
M
on
A
C
AC
A
C
and
B
C
BC
BC
. Suppose that the incenter and circumcenter of the triangle lie on the segment
K
L
KL
K
L
.(a) Prove that
C
D
=
R
+
r
CD=R+r
C
D
=
R
+
r
, where
R
R
R
and
r
r
r
are the circumradius and inradius, respectively. (b) Find the minimum value of the ratio
C
M
:
C
D
CM:CD
CM
:
C
D
.
geometry
Triangles