positive integer arrays 
,
,
![\[
\sqrt[3]{\sqrt[3]{x - \frac{3}{8}} - \frac{3}{8}} = x^3 + \frac{3}{8}
\]](http://latex.artofproblemsolving.com/b/b/9/bb99cd588b94b5716738e6e9ea01d15377b6efef.png)
has exactly two real positive solutions 
and 
.
.
and 
Prove that
,
and 
meet at a point 
.
and the points 
lie on the same straight line.
Y by
My question is after the problem and my solution.
Problem:
In the quadrilateral
, 
, 
, 
.
(1) Find
.
(2) If
, find 
. (2018 China Gaokao Syllabus I #17)
Solution:
Draw a diagram (see the attachment):
(1) First,
. Since 
, we can plug it back and find that 
. Therefore, through the Pythagorean Identities, we find that 
.
(2) Notice that since, its sine is 
. Therefore, using the fact that 
, we can say: 
.
Expand using sum of sines:
.
Using
, call 
equal to 
(notice that since 
is acute, its sine must be positive, so that's why I used the cosine to set the value to avoid problems), 
. Plug it back in and solve the quadratic equation, we find that 
. Now using the law of cosines, we find that 
.
Wow! Such a long bash! Thanks for reading the solution. Now, here's the question:
In the answer key to part (2) of this problem, it said:
Because of the problem's given information and part (1) [its solution is identical to mine, just differently worded], it is obvious that
.
How come? I bashed all of this out to obtain that fact. How come it's "obvious?" Please explain. Notice that if the answer key omitted these details, it wouldn't be an answer key because the bashing is vital to this problem it looks like!
Thanks!
Problem:
In the quadrilateral
, 
, 
, 
.(1) Find
.(2) If
, find 
. (2018 China Gaokao Syllabus I #17)Solution:
Draw a diagram (see the attachment):
(1) First,
. Since 
, we can plug it back and find that 
. Therefore, through the Pythagorean Identities, we find that 
.(2) Notice that since
, its sine is 
. Therefore, using the fact that 
, we can say: 
.Expand using sum of sines:
.Using
, call 
equal to 
(notice that since 
is acute, its sine must be positive, so that's why I used the cosine to set the value to avoid problems), 
. Plug it back in and solve the quadratic equation, we find that 
. Now using the law of cosines, we find that 
.Wow! Such a long bash! Thanks for reading the solution. Now, here's the question:
In the answer key to part (2) of this problem, it said:
Because of the problem's given information and part (1) [its solution is identical to mine, just differently worded], it is obvious that
. How come? I bashed all of this out to obtain that fact. How come it's "obvious?" Please explain. Notice that if the answer key omitted these details, it wouldn't be an answer key because the bashing is vital to this problem it looks like!
Thanks!
Attachments:
So the fact that 
is a direct result from the fact that 
The proof of this isn't too hard. Just consider the unit circle. Notice that rotating the point 
by an angle of 
counterclockwise takes it to the point 
and rotating it 
Also notice that these two points are symmetric about 
so the points 
and 
are the same, so 
and 
