256. Find the radius of the circle circumscribed around a triangle, given its sides.

## Task of Campano of Navarre.

From a comment on the Latin translation from Arabic of Euclid's "Elements".

##
256. Let the sides of the triangle be \( AC = b, BC = a, AB = c \) (Fig. 59). The height \( h = BE \) is unknown. By drawing the diameter \( BD \), we have from the similarity of triangles \( ABD \) and \( EEC \):

![](https://cdn.mathpix.com/cropped/2024_05_21_926f0bb262c8f9569516g-136.jpg?height=411&width=417&top_left_y=1459&top_left_x=383)

Fig. 59.

\[
a \cdot c = 2R \cdot h
\]

where \( R \) is the desired radius.

Knowing the three sides, we can calculate the area \( \Delta \) of the triangle. But then

\[
\frac{b h}{2} = \Delta
\]

from which

\[
h = \frac{2 \Delta}{b}
\]

Therefore, substitution gives \( a c = \frac{4 R \Delta}{b} \), i.e., \( R = \frac{a c b}{4 \Delta} \).

Campanus of Novara (late 13th century) is known for his Latin translation of Euclid's "Elements" from Arabic, along with additions, i.e., Books XIV and XV, the first of which is attributed to Hypsicles, and the second to Damascius. Thanks to this work, which was published with commentaries in 1482, knowledge of Euclid spread throughout Europe. Regarding problem 257, Campanus states that this proposition was already known to Aristaeus and Apollonius.