Conditions which must exist in order that two circles may intersect.
Properties of the secants of the circle.
Two secants which start from the same point without the circle, being prolonged to the most distant part of the circumference, are reciprocally proportional to their exterior segments.—The tangent is a mean proportional between the secant and its exterior segment.
Two chords intersecting within a circle divide each other into parts reciprocally proportional.—The line perpendicular to a diameter and terminated by the circumference, is a mean proportional between the two segments of the diameter.
A chord, passing through the extremity of the diameter, is a mean proportional between the diameter and the segment formed by the perpendicular let fall from the other extremity of that chord.—To find a mean proportional between two given lines.
To divide a line in extreme and mean ratio.—The length of the line being given numerically, to calculate the numerical value of each of the segments.
Of polygons inscribed and circumscribed to the circle.
To inscribe or circumscribe a circle to a given triangle.
Every regular polygon can be inscribed and circumscribed to the circle.
A regular polygon being inscribed in a circle, 1o to inscribe in the same circle a polygon of twice as many sides, and to find the length of one of the sides of the second polygon; 2o to circumscribe about the circle a regular polygon of the same number of sides, and to express the side of the circumscribed polygon by means of the side of the corresponding inscribed polygon.