from its extremity; the line intercepted between the vertex and the
latter perpendicular is equal to the radius of the circumscribing
circle.
10. If a triangle be inscribed in a semicircle, and a perpen-
dicular drawn from any point in the diameter meeting one side, the
circumference, and the other side produced; the segments cut off
will be in continued proportion.
11. If a triangle be inscribed in a semicircle, and one side be
equal to the semidiameter; the other side will be a mean proportional
between that side and a line equal to that side and the diameter
together.
12. If a circle be inscribed in a right-angled triangle; to deter-
mine the least angle that can be formed by two lines drawn from the
extremity of the hypothenuse to the circumference of the circle.
13. If an equilateral triangle be inscribed in a circle, and through
the angular points another be circumscribed; to determine the ratio
which they bear to each other.
14. A straight line drawn from the vertex of an equilateral
triangle inscribed in a circle to any point in the opposite circum-
ference, is equal to the two lines together, which are drawn from
the extremities of the base to the same point.
15. If the base of a triangle be produced both ways, so that each
part produced may be equal to the adjacent side, and through the
extremities of the parts produced and the vertex a circle be de-
scribed; the line joining its centre and the vertex of the triangle
will bisect the angle at the vertex.
16. If an isosceles triangle be inscribed in a circle, and from the
vertical angle a line be drawn meeting the circumference and the
base; either equal side is a mean proportional between the segments
of the line thus drawn.
17. If from the extremities of one of the equal sides of an
isosceles triangle inscribed in a circle, tangents be drawn to the circle
and produced to meet; two lines drawn to any point in the circum-
ference from the point of concourse and one point of contact, will
divide the base (produced if necessary) in geometrical proportion.
18. If on the sides of a triangle segments of circles be described