the rectangle contained by the segments AG, AH of the diameter, is equivalent to the square of the tangent AB.

PROP. XXVII. PROB.

To construct a square equivalent to a given rectilineal figure.

A

B

Let the rectilineal figure be reduced by Proposition 6. Book II. to an equivalent rectangle, of which A and B are the two containing sides; draw an indefinite straight line CE, in which take the part CD equal to A and DE to B, on C describe a semicircle, and erect the perpendicular DF from the diameter to meet the circumference: DF is the side of the square equivalent to the given rectilineal figure.

C

D

E

For, by Cor. 1. to the last Proposition, the square of the perpendicular DF is equivalent to the rectangle under the segments CD, DE of the diameter, and is consequently equivalent to the rectangle contained by the sides A and B of a rectangle that was made equivalent to the rectilineal figure.

PROP. XXVIII. THEOR.

A quadrilateral figure may have a circle described about it, if the rectangles under the segments made by the intersection of its diagonals be equi

valent, or if those rectangles are equivalent which are contained by the external segments formed by producing its opposite sides.

Let ABCD be a quadrilateral figure, of which AC and BD are the diagonals, and such that the rectangle AE, EC is equivalent to the rectangle BE, ED; a circle may be made to pass through the four points A, B, C, and D. For describe a circle through

the three points A, B, C (III. 9. cor.), and let it cut BD in G. Because AC and BG intersect each other within a circle, the rectangle AE, EC is equivalent to the rectangle BE, EG (III. 26.); but

F

B

E

the rectangle AE, EC is by hypothesis equivalent to the rectangle BE, ED. Wherefore BE, EG is equivalent to BE, ED; and these rectangles have a common base BE, consequently (II. 3. cor.) their altitudes EG and ED are equal, and hence the point G is the same as D, or the circle passes through all the four points A, B, C, and D.

Again, if the opposite sides CB and DA be produced to meet at F, and the rectangle CF, FB be equal to DF, FA, a circle may be described about the figure.

For, as before, let a circle pass through the three points A, B, C, but cut AD in H. And from the property of the circle, the rectangle CF, FB is equivalent to HF, FA; but the rectangle CF, FB is also equivalent to DF, FA; whence the rectangle HF, FA is equivalent to DF, FA, and the base HF equal to DF, or the point H is the same as D.

ELEMENTS

OF

GEOMETRY.

BOOK IV.

DEFINITIONS.

1. A rectilineal figure is said to be inscribed in a circle, when all its angular points lie on the circumference.

2. A rectilineal figure circumscribes a circle, when each of its sides is a tangent.

3. A circle is inscribed in a rectilineal figure, when it touches all the sides.

4. A circle is described about a rectilineal figure or circumscribes it, when the circumference passes through all the angular points of the figure.

5. Polygons are equilateral, when their sides, in the same order, are respectively equal: They are equiangular, if an equality obtains between their corresponding angles.

6. Polygons are said to be regular, when all their sides and their angles are equal.