,
,
,
,
or even expressions of those terms combined as if that would make them defined. I have made this post to answer these questions once and for all, and I politely ask everyone to link this post to threads that are talking about this issue.
,
isn't even rigorously defined in this expression. What exactly do we mean by 
.
,
does not exist in the first place, although you will see why in a calculus course. So the point is that 
?
indeterminate? In analysis (the theory behind calculus and beyond), limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits. However, if the expression obtained after this substitution does not provide sufficient information to determine the original limit, then the expression is called an indeterminate form. For example, we could say that 
,
or ![\[0/0, \infty/\infty, \infty-\infty, \infty \times 0\]](http://latex.artofproblemsolving.com/2/6/0/260f97a1cf1ea8722ff97a29b83bac7bfe83e577.png)
etc etc. So what makes something undefined? In the broader scope, something being undefined refers to an expression which is not assigned an interpretation or a value. A function is said to be undefined for points outside its domain. For example, the function 
given by the mapping 
is undefined for 
.
is undefined because dividing by ![\[\overline{\mathbb{R}}=\mathbb{R}\cup \{-\infty,+\infty\}.\]](http://latex.artofproblemsolving.com/9/9/3/993c78b62ec2c84b5d77daa8e9a8cd6f70fcf904.png)
So instead of treating infinity as an idea, we define infinity (positively and negatively, mind you) as actual numbers in the reals. The advantage of doing this is for two reasons. The first is because we can turn this thing into a totally ordered set. Specifically, we can let 
for each 
which means that via this order topology each subset has an infimum and supremum and 
is therefore compact. While this is nice from an analytic standpoint, extending the reals in this way can allow for interesting arithmetic! In ![\begin{align*}
a + \infty = \infty + a & = \infty, & a & \neq -\infty \\
a - \infty = -\infty + a & = -\infty, & a & \neq \infty \\
a \cdot (\pm\infty) = \pm\infty \cdot a & = \pm\infty, & a & \in (0, +\infty] \\
a \cdot (\pm\infty) = \pm\infty \cdot a & = \mp\infty, & a & \in [-\infty, 0) \\
\frac{a}{\pm\infty} & = 0, & a & \in \mathbb{R} \\
\frac{\pm\infty}{a} & = \pm\infty, & a & \in (0, +\infty) \\
\frac{\pm\infty}{a} & = \mp\infty, & a & \in (-\infty, 0).
\end{align*}](http://latex.artofproblemsolving.com/6/d/8/6d8fc1eca5c791d430a7168cd3b37a02104fa318.png)
So addition, multiplication, and division are all defined nicely. However, notice that we have some indeterminate forms here which are also undefined,![\[\infty-\infty,\frac{\pm\infty}{\pm\infty},\frac{\pm\infty}{0},0\cdot \pm\infty.\]](http://latex.artofproblemsolving.com/a/0/b/a0b67076efcc014c78b2023be1151c84b57c1503.png)
So while we define certain things, we also left others undefined/indeterminate in the process! However, in the context of measure theory it is common to define 
as greenturtle3141 noted below. I encourage to reread what he wrote, it's great stuff! As you may notice, though, dividing by ![\[\mathbb{C}^*=\mathbb{C}\cup\{\tilde{\infty}\}\]](http://latex.artofproblemsolving.com/8/4/e/84ed3d6ab237df970333b87beaef5311caa46185.png)
which we call the Riemann Sphere (it actually forms a sphere, pretty cool right?). As a note, 
means complex infinity, since we are in the complex plane now. Here's the catch: division by ![\[\frac{z}{0}=\tilde{\infty},\frac{z}{\tilde{\infty}}=0.\]](http://latex.artofproblemsolving.com/1/6/0/160cb939707708ff4e0930724df2928af11d7fd1.png)
where 
and 
Furthermore, we actually have some nice properties with multiplication that we didn't have before. In 
it holds that![\[\tilde{\infty}\times \tilde{\infty}=\tilde{\infty}\]](http://latex.artofproblemsolving.com/8/e/0/8e035b4841d20c5ad671c46355cc667ca01e533b.png)
but 
and 
are left as undefined (unless there is an explicit need to change that somehow). One could define the projectively extended reals as we did with ![\[{\widehat {\mathbb {R} }}=\mathbb {R} \cup \{\infty \}.\]](http://latex.artofproblemsolving.com/c/0/4/c044ff37849acad3b8b280b9753539225a1276ca.png)
They behave in a similar way to the Riemann Sphere, with division by 
be the set of all positive integers. Determine all subsets 
of 
such that
and 
intersect at 
and 
.
;
and 
,
and 
intersects 
.
on 
.
and 
meet at 
,
.
that satisfy 
,
?
.
Y by Adventure10
Let 
be a triangle and 
be its circumcenter. Let 
be two points in the interior of , 
the reflection of wrt of the midpoint of 
, and 
the reflection of wrt of the Euler circle center. Let 
be the circle that passes through such that and the circumcenter are orthogonal. Let 
intersect in 
and 
. Let 
intersect at 
. Prove that if are isogonal conjugates, then 
are also isogonal conjugates.
be a triangle and 
be its circumcenter. Let 
be two points in the interior of , 
the reflection of wrt of the midpoint of 
, and 
the reflection of wrt of the Euler circle center. Let 
be the circle that passes through such that and the circumcenter are orthogonal. Let 
intersect in 
and 
. Let 
intersect at 
. Prove that if are isogonal conjugates, then 
are also isogonal conjugates.
with isogonal conjugate 
Let 
be the midpoint of 
and let 
be the image of 
under the inversion WRT 
respectively. Prove that the isogonal conjugate of the anticomplement of 
be the second intersection of 
with 
respectively and let 
be the midpoint of 
First, recall the following well-known properties :
and the intersection of 
lies on 
of 
Let 
be the midpoint of 
be the tangency point of 
since ![$\color{blue} ^{[1]}$](http://latex.artofproblemsolving.com/c/2/c/c2c92fce3d7ab9715fa3c9ac2923e39a591b7542.png)
is parallel to the polar of 
WRT 
respectively, hence 
is parallel to 
the anticomplement 
of 
be the isogonal conjugate of 
be the reflection of 
so by DDIT for 
and quadrilateral 
we get 
are in an involution, where 
is the orthocenter of 
Note that 
pass through the antigonal conjugate of 
respectively, so reflect these pairs in the bisector of 
yields 
are in an involution, where 
is the circumcenter of 
is the intersection of 
then by DDIT for 
we get 
Similarly, we can prove that 
so 
i.e. the isogonal conjugate of 
