Proof: By Euclid
(related to Proposition: 4.05: Circumscribing Circle about Triangle)
 Let the straight lines $AB$ and $AC$ have been cut in half at points $D$ and $E$ (respectively) [Prop. 1.10].
 And let $DF$ and $EF$ have been drawn from points $D$ and $E$, at right angles to $AB$ and $AC$ (respectively) [Prop. 1.11].
 So ($DF$ and $EF$) will surely either meet inside triangle $ABC$, on the straight line $BC$, or beyond $BC$.
 Let them, first of all, meet inside (triangle $ABC$) at (point) $F$, and let $FB$, $FC$, and $FA$ have been joined.
 And since $AD$ is equal to $DB$, and $DF$ is common and at right angles, the base $AF$ is thus equal to the base $FB$ [Prop. 1.4].
 So, similarly, we can show that $CF$ is also equal to $AF$.
 So that $FB$ is also equal to $FC$.
 Thus, the three (straight lines) $FA$, $FB$, and $FC$ are equal to one another.
 Thus, the circle drawn with center $F$, and radius one of $A$, $B$, or $C$, will also go through the remaining points.
 And the circle will have been circumscribed about triangle $ABC$.
 Let it have been (so) circumscribed, like $ABC$ (in the first diagram from the left).
 And so, let $DF$ and $EF$ meet on the straight line $BC$ at (point) $F$, like in the second diagram (from the left).
 And let $AF$ have been joined.
 So, similarly, we can show that point $F$ is the center of the circle circumscribed about triangle $ABC$.
 And so, let $DF$ and $EF$ meet outside triangle $ABC$, again at (point) $F$, like in the third diagram (from the left).
 And let $AF$, $BF$, and $CF$ have been joined.
 And, again, since $AD$ is equal to $DB$, and $DF$ is common and at right angles, the base $AF$ is thus equal to the base $BF$ [Prop. 1.4].
 So, similarly, we can show that $CF$ is also equal to $AF$.
 So that $BF$ is also equal to $FC$.
 Thus, [again] the circle drawn with center $F$, and radius one of $FA$, $FB$, and $FC$, will also go through the remaining points.
 And it will have been circumscribed about triangle $ABC$.
 Thus, a circle has been circumscribed about the given triangle.
 (Which is) the very thing it was required to do.
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References
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"