Inscribing a Circle in a Triangle

Theorem

In any given triangle it is possible to inscribe a circle.

Construction

Let $ABC$ be the given triangle.

Let $\angle ABC$ and $\angle ACB$ be bisected by $BD$ and $CD$ and let these lines join at $D$.

From $D$ construct the perpendiculars $DE, DF, DG$ to $AB, BC, AC$ respectively.

Draw the circle with radius $DE$ and center $D$.

This is the required circle.

Proof

We have that $\angle ABD = \angle CBD$ and $\angle BED = \angle BFD$, a right angle, and $BD$ is common.

So from Triangle Angle-Side-Angle and Side-Angle-Angle Equality we have that $\triangle EBD = \triangle FBD$.

So $DE = DF$.

For the same reason $DG = DF$.

So $DE = DF = DG$.

So the circle drawn with radius $DE$ will pass through $E, F$ and $G$.

From Line at Right Angles to Diameter of Circle it follows that $AB, AC, BC$ are tangent to the circle $EFG$.

Hence the result.

$\blacksquare$

Historical Note

This is Proposition 4 of Book IV of Euclid's The Elements.