Angle oriented geometry
by Problems_eater, Apr 15, 2025, 9:03 PM
Let 
be four distinct points in the plane.
Which of the following statements, expressed using oriented angles, are always true?
1.If lines
and 
are distinct and parallel, then
the oriented angle
is equal to the oriented angle DCB.
2.If
lies on the segment 
, then
the oriented angle
plus the oriented angle 
equals 
.
3.If the oriented angle
plus the oriented angle 
equals 0°, then
lines
and are parallel.
4.If the oriented angle plus the oriented angle equals , then
lines and are parallel.
be four distinct points in the plane.Which of the following statements, expressed using oriented angles, are always true?
1.If lines
and 
are distinct and parallel, thenthe oriented angle
is equal to the oriented angle DCB.2.If
lies on the segment 
, thenthe oriented angle
plus the oriented angle 
equals 
.3.If the oriented angle
plus the oriented angle 
equals 0°, thenlines
and are parallel.4.If the oriented angle
plus the oriented angle equals , thenlines
and are parallel.
cannot completely cover a square of side length 
,
be reals such that 
. Prove that
![\[
\frac{(x+y+z)^5 - x^5 - y^5 - z^5}{(x+y)(y+z)(z+x)} = 0
\]](http://latex.artofproblemsolving.com/7/8/f/78f8e6c2570684631dc288144e13568d71e68ec8.png)
and 
Let 
be a point on that plane such that 
Prove that ![$$AP \in \left[\frac{5-\sqrt{5}}{10}, \frac{-1+\sqrt{5}}{2}\right] \cup \left[\frac{5+\sqrt{5}}{10}, \frac{1+\sqrt{5}}{2}\right].$$](http://latex.artofproblemsolving.com/7/8/e/78e428cbc99ce4e9448f5c4cec86f50567123812.png)
specifies the greatest common divisor of 
and 
.
,
.
to meet the circumcircle of 
at 
,
.
and 
Prove that
Let 
Prove that
September 2024
June 2024