Circumscribing Circle about Triangle

Theorem

In the words of Euclid:

About a given triangle to circumscribe a circle.

(The Elements: Book $\text{IV}$: Proposition $5$)

Let $\triangle ABC$ be the given triangle.

Let the straight lines $AB$ and $AC$ be bisected at $D$ and $E$.

Let $DF$ and $EF$ be drawn perpendicular to $AB$ and $AC$ respectively.

Let a circle be drawn with center $F$ and radius $AF$.

This is the circle required.

The point $F$ will be either inside, outside or on the edge $BC$ of $\triangle ABC$.

First suppose $F$ is inside $\triangle ABC$.

Join $FB$ and $FC$.

We have that $AD = DB$ and $DF$ is common and at right angles to $AB$.

So from Triangle Side-Angle-Side Congruence, $\triangle ADF = \triangle BDF$, and so $AF = BF$.

Similarly we can prove that $BF = CF$.

So $AF = BF = CF$.

Thus the circle with center $F$ and radius $AF$ also passes through $B$ and $C$.

That is, it circumscribes $\triangle ABC$.

$\Box$

Secondly, suppose $F$ lies on $BC$.

We have that $AD = DB$ and $DF$ is common and at right angles to $AB$.

So from Triangle Side-Angle-Side Congruence, $\triangle ADF = \triangle BDF$, and so $AF = BF$.

Similarly we can prove that $BF = CF$.

So $AF = BF = CF$.

Thus the circle with center $F$ and radius $AF$ also passes through $B$ and $C$.

That is, it circumscribes $\triangle ABC$.

$\Box$

Thirdly, suppose $F$ lies outside $\triangle ABC$.

Join $FB$ and $FC$.

We have that $AD = DB$ and $DF$ is common and at right angles to $AB$.

So from Triangle Side-Angle-Side Congruence, $\triangle ADF = \triangle BDF$, and so $AF = BF$.

Similarly we can prove that $BF = CF$.

So $AF = BF = CF$.

Thus the circle with center $F$ and radius $AF$ also passes through $B$ and $C$.

That is, it circumscribes $\triangle ABC$.

$\blacksquare$

This proof is Proposition $5$ of Book $\text{IV}$ of Euclid's The Elements.

1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 2 (2nd ed.) ... (previous) ... (next): Book $\text{IV}$. Propositions
1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {IV}$. Pure Geometry: Plane Geometry: The circumcentre
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): circumcentre
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): circumcentre (circumcircle)