CASE 2-When A is out of the Circumference. From the given point A BA draw Ao to the centre, which bisect in the point m. With the centre m, and radius mA or mo, describe an arc, cutting the given circle inn. Through the points A and n draw the tangent B C. PROBLEM XV. To find a third Proportional to two given Lines AB, AC. Place the two given lines, AB, A. A C, making any angle at A, A. and join BC. In AB take AD equal to AC, and draw DE parallel to B C. So shall AE be the third proportional to AB and A C. That is, AB: AC :: AC: AE. PROBLEM XVI. E C B C D a To find a fourth Proportional to three given Lines AB, AC, AD. A Place two of them, AB, AC, A so as to make any angle at A, and A join B C. Place AD on AB, and draw D E parallel to BC. So shall A E be the fourth proportional re quired. That is, AB: AC :: AD: ΑΕ. PROBLEM XVII, To find a mean Proportional between two given Lines AB, ВС. To divide a Line AB in Extreme and Mean Ratio. PROBLEM XIX. To inscribe an isosceles triangle in a given circle, that shall have each of the angles at the base double the angle at the vertex. Draw any diameter A B of the given circle; and divide the radius CB, in the point D, in extreme and mean ratio, by the last problem. From the point B apply the chords BE, BF, each equal to CD; then join AE, AF, EF, and AEF will be the triangle required. PROBLEM XX. To make an equilateral Triangle on a given Line AB. NOTE. An isosceles triangle may be made in the same manner, by taking for the radius the given length of one of the equal sides. PROBLEM PROBLEM XXI. To make a Triangle with three given Lines AB, AС, В С. With the centre A and radius PROBLEM XXII. To make a Square upon a given Line AB. PROBLEM XXIII. To describe a Rectangle, or a Parallelogram, of a given Length and Breadth. NOTE. In the same manner is described any oblique parallelogram, except in drawing BC so as to make the given oblique angle with AB, instead of a right one. : r To make a regular Pentagon on a given Line AB. Make B im perpendicular and equal to half AB. Draw Am, and produce it till m n be equal to B m. With centres A and B, and radius Bn, describe arcs intersecting in o, which will be the centre of the cir cumscribing circle. Then with A the centre o, and the same rad : B 32 ius, describe the circle; and about the circumference of it apply A B the proper number of times. Another |