MathDB
Problems
Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
2005 Czech And Slovak Olympiad III A
3
3
Part of
2005 Czech And Slovak Olympiad III A
Problems
(1)
tangent quadrilaterals ABED and AECD in trapezoid ABCD
Source: Czech and Slovak MO, III A, 2005 p3
1/12/2020
In a trapezoid
A
B
C
D
ABCD
A
BC
D
with
A
B
/
/
C
D
,
E
AB // CD, E
A
B
//
C
D
,
E
is the midpoint of
B
C
BC
BC
. Prove that if the quadrilaterals
A
B
E
D
ABED
A
BE
D
and
A
E
C
D
AECD
A
EC
D
are tangent, then the sides
a
=
A
B
,
b
=
B
C
,
c
=
C
D
,
d
=
D
A
a = AB, b = BC, c =CD, d = DA
a
=
A
B
,
b
=
BC
,
c
=
C
D
,
d
=
D
A
of the trapezoid satisfy the equalities
a
+
c
=
b
3
+
d
a+c = \frac{b}{3} +d
a
+
c
=
3
b
+
d
and
1
a
+
1
c
=
3
b
\frac1a +\frac1c = \frac3b
a
1
+
c
1
=
b
3
.
geometry
trapezoid
tangent