Inscribing Circle in Triangle

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In the words of Euclid:

In a given triangle to inscribe a circle.

(The Elements: Book $\text{IV}$: Proposition $4$)



Let $\triangle ABC$ 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.


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 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.


Historical Note

This proof is Proposition $4$ of Book $\text{IV}$ of Euclid's The Elements.