Task
The goal of this task is to show how to draw a circle which is tangent to
all three sides of a given triangle: that is, the circle touches each side of the
triangle in a single point.
Suppose $ABC$ is a triangle as pictured below with ray $\overrightarrow{AO}$ the bisector
of angle $A$ and ray $\overrightarrow{BO}$ the bisector of angle $B$:
Also pictured are the point $M$ on $\overline{AC}$ so that $\overline{OM}$ meets
$\overline{AC}$ in a right angle and similarly point $N$ on $\overline{AB}$ is chosen so
that $\overline{ON}$ meets $\overline{AB}$ in a right angle.

Show that triangle $AOM$ is congruent to triangle $AON$.

Show that $\overline{OM}$ is congruent to segment $\overline{ON}$.

Arguing as in parts (a) and (b) show that if $P$ is the point on $\overline{BC}$
so that $\overline{OP}$ meets $\overline{BC}$ in a right angle then $OP$ is also congruent to $\overline{OM}$.

Show that the circle with center $O$ and radius $OM$ is inscribed inside
triangle $ABC$.
IM Commentary
This task shows how to inscribe a circle in a triangle using angle
bisectors. A companion task, ''Inscribing a circle in a triangle II'' stresses
the auxiliary remarkable fact that comes out of this task, namely that the
three angle bisectors of triangle $ABC$ all meet in the point $O$. In order
to complete part (d) of this task, students need to know that the tangent line
to a circle at a point $p$ is characterized by being perpendicular to the radius of
the circle at $p$.
This task is primarily intended for instruction purposes but parts (a) and (b)
could be used for assessment. The teacher is encouraged to draw many different
triangles both to provide a broader context for the problem and to see where the
inscribed circle lies depending on the shape of the triangle. If students
no how to bisect an angle and draw the segment from a point to a line, meeting
the line in a right angle, then they could perform the entire construction with straightedge and compass. This particular construction has a great reward at the end, namely being able to produce the inscribed circle inside the triangle.