An Angle Invariant for Arbitrary Triangles

​
cosϕ
S
A
xu+
S
B
yv+
S
C
zw

S
A
2
x
+
S
B
2
y
+
S
C
2
z

S
A
2
u
+
S
B
2
v
+
S
C
2
w

-0.21685
-0.21685
Let
ϕ
be the angle between two arbitrary lines
EF
and
GH
. Let
ΔABC
be an arbitrary triangle.
Define values relating to quadrilaterals based on the line
EF
:
x=
2
EB
+
2
FC
-
2
EC
-
2
FB
,
y=
2
EC
+
2
FA
-
2
EA
-
2
FC
,
z=
2
EA
+
2
FB
-
2
EB
-
2
FA
.
Define values relating to quadrilaterals based on the line
GH
:
u=
2
GB
+
2
HC
-
2
GC
-
2
HB
,
v=
2
GC
+
2
HA
-
2
GA
-
2
HC
,
w=
2
GA
+
2
HB
-
2
GB
-
2
HA
.
Then:
cosϕ=±
S
A
xu+
S
B
yv+
S
C
zw

S
A
2
x
+
S
B
2
y
+
S
C
2
z

S
A
2
u
+
S
B
2
v
+
S
C
2
w

,
where
S
A
,
S
B
,
S
C
are Conway notation for the triangle
ABC
.
You can drag the points.

Permanent Citation

Minh Trinh Xuan
​
​"An Angle Invariant for Arbitrary Triangles"​
​http://demonstrations.wolfram.com/AnAngleInvariantForArbitraryTriangles/​
​Wolfram Demonstrations Project​
​Published: May 16, 2022