Incenter and Incircle

It has a triangle, and triangles are beautiful.

It has a circle, and circles are beautiful.

The circle just touches the triangle on each of its sides.

Shift the circle over a bit until it just touches two of the sides

Give up

It has the same distance to lines $AB$ and $AC$

It has the same distance to lines $AB$ and $BC$

It has the same distance to points $B$ and $C$.

$A$

$B$

$C$

$F$

Expand (dilate) it about the current point $F$

Adjust its size, keeping it tangent to lines $AB$ and $AC$, until it touches $BC$ too.

$A$

$B$

$C$

$F$

$I$

The median from $A$

The perpendicular bisector of side $BC$

The symmedian from $A$

The angle bisector of $\angle B A C$

Yes!

No, it fails for obtuse triangles.

No, it fails for equilateral triangles.

Medians

Symmedians

Angle bisectors

Perpendicular bisectors

They always intersect at some point, which we could call $I$.

They don't always intersect; there are some weird triangles out there.

$I$ has the same distance to points $A$ and $C$.

$I$ has the same distance to lines $AB$ and $AC$.

$I$ has the same distance to lines $CA$ and $CB$.

$I$ has the same distance to all three sides of triangle $ABC$.

$I$ is also on the angle bisector of $\angle C B A$

All three angle bisectors of triangle $ABC$ intersect at a single point, and that point is $I$.

The intersection of perpendicular bisectors of the sides of $\bigtriangleup ABC$

The intersection of the internal angle bisectors of $\bigtriangleup ABC$

The incircle of $\bigtriangleup ABC$ is tangent to each of the sides of $\bigtriangleup ABC$

The intersection of medians