§ 40. Of somewhat greater interest are the next problems, where the triangles are given and the circles to be found.
Prop. 4. To inscribe a circle in a given triangle.
The result is that the problem has always a solution, viz. the centre of the circle is the point where the bisectors of two of the interior angles of the triangle meet. The solution shows, though Euclid does not state this, that the problem has but one solution; and also,
The three bisectors of the interior angles of any triangle meet in a point, and this is the centre of the circle inscribed in the triangle.
The solutions of most of the other problems contain also theorems. Of these we shall state those which are of special interest; Euclid does not state any one of them.
§ 41. Prop. 5. To circumscribe a circle about a given triangle.
The one solution which always exists contains the following:—
The three straight lines which bisect the sides of a triangle at right angles meet in a point, and this point is the centre of the circle circumscribed about the triangle.
Euclid adds in a corollary the following property:—
The centre of the circle circumscribed about a triangle lies within, on a side of, or without the triangle, according as the triangle is acute-angled, right-angled or obtuse-angled.