MathDB
Problems
Contests
National and Regional Contests
Latvia Contests
Latvia BW TST
2018 Latvia Baltic Way TST
P11
P11
Part of
2018 Latvia Baltic Way TST
Problems
(1)
Given angles of triangle, prove third order segment equality
Source: 2018 Latvia BW TST P11
3/26/2022
Let
A
B
C
ABC
A
BC
be a triangle with angles
∠
A
=
8
0
∘
,
∠
B
=
7
0
∘
,
∠
C
=
3
0
∘
\angle A = 80^\circ, \angle B = 70^\circ, \angle C = 30^\circ
∠
A
=
8
0
∘
,
∠
B
=
7
0
∘
,
∠
C
=
3
0
∘
. Let
P
P
P
be a point on the bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
satisfying
∠
B
P
C
=
13
0
∘
\angle BPC =130^\circ
∠
BPC
=
13
0
∘
. Let
P
X
,
P
Y
,
P
Z
PX, PY, PZ
PX
,
P
Y
,
PZ
be the perpendiculars drawn from
P
P
P
to the sides
B
C
,
A
C
,
A
B
BC, AC, AB
BC
,
A
C
,
A
B
, respectively. Prove that the following equation with segment lengths is satisfied
A
Y
3
+
B
Z
3
+
C
X
3
=
A
Z
3
+
B
X
3
+
C
Y
3
.
AY^3+BZ^3+CX^3=AZ^3+BX^3+CY^3.
A
Y
3
+
B
Z
3
+
C
X
3
=
A
Z
3
+
B
X
3
+
C
Y
3
.
geometry
angle bisector