Circle Inscribed in a Triangle

In this video you will learn about the properties of the line segments formed when a circle is inscribed in a triangle.

In this video, we are going to look at a circle inscribed in a triangle.
Here we have triangle ABC and a circle inscribed in it that is tangent at points $D$ , $E$ , and $F$ . These tangent points are NOT midpoints of the sides of the triangle. Line segment $\overline{AD}$ is not the same length as line segment $\overline{DB}$ . However, something does happen where some line segments are equal. If we look at the triangle on the side and draw the tangent lines, what happens is that the two tangent lines are congruent to each other. So:
$\overline{BD}\cong\overline{BE}$
$\overline{AD}\cong\overline{AF}$
$\overline{FC}\cong\overline{EC}$
If we were given a few measurements, it would be very possible to find many other measurements.
Given that $\overline{BD}$ is 3, $\overline{AC}$ is 12, and $\overline{EC}$ is 4, let’s see what we can do.
Since $\overline{BD}$ is 3, then $\overline{BE}$ is also 3. This means that the whole length of $\overline{BC}$ is 7.
Since $\overline{EC}$ is 4, then $\overline{FC}$ is also 4. This means that $\overline{AF}$ must be 8.
Since $\overline{AF}$ is 8, then $\overline{AD}$ is also 8. This means that the whole length of $\overline{AB}$ is 11.