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To construct a square equal to a given triangle.
Let ABC be the given triangle, AD its altitude, and BC its base.
square then will this be the square required. For, from the construction,
XY2 ¿BC × AD = area ABC.
Scholium. By means of Problems VI. and VII., a square may be constructed equal to any given polygon.
On a given line, to construct a polygon similar to a given
Let FG be the given line, and ABCDE the given polygon. Draw AC and AD.
in like manner, construct the triangle FHI similar to ACD, and FIK similar to ADE; then will the polygon FGHIK be similar to the polygon ABCDE (P. XXVI., C.).
To construct a square equal to the sum of two given squares, also a square equal to the difference of two given squares.
1o. Let A and B be the sides of the given squares, and let ▲ be the greater.
Construct a right angle CDE; make DE equal to A, and DC equal to
B; draw CE, and on it
construct a square: this square will be equal to the sum
of the given squares (P. XI.).
2o. Construct a right angle CDE
Lay off DC equal to B;
with C CE, equal to
as a centre, and
Scholium. By means of Probs. VI., VII., VIII., and IX., a polygon may be constructed similar to two given polygons, and equal to their sum, or to their difference (P. XXVII, C.).
REGULAR POLYGONS.—AREA OF THE CIRCLE.
1. A REGULAR POLYGON is a polygon which is both equilateral and equiangular.
PROPOSITION I. THEOREM.
Regular polygons of the same number of sides are similar.
Let ABCDEF and abcdef be regular polygons of the same number of sides: then will they be similar.
vided by the number of angles (B. I., P. XXVI., C. 4); and further, the corresponding sides are proportional, because all the sides of either polygon are equal (D. 1) hence, the polygons are similar (B. IV., D. 1); which was to be proved.
PROPOSITION II. THEOREM.
The circumference of a circle may be circumscribed about any regular polygon; a circle may also be inscribed within it.
1o. Let ABCF be a regular polygon: then can the circumference of a circle be circumscribed about it.
For, through three consecutive ver
tices A, B, C, describe the circumference of a circle (B. III., Problem XIII., S.). Its centre O will lie on PO, drawn perpendicular to BC, at its middle point P; draw OA and OD.
Let the quadrilateral OPCD be turned about the line OP, until PC falls on PB; then, because the angle C is equal to B, the side CD will take the direction BA; and because CD is equal to BA, the vertex D, will fall upon the vertex A; and consequently, the line OD will coincide with OA, and is, therefore, equal to it: hence, the circumference which passes through A, B, and C, will pass through D. In like manner, it may be shown that it will pass through all of the other vertices: hence, it is circumscribed about the polygon; which was to be proved.
2o. A circle may be inscribed within the polygon.
For, the sides AB, BC, &c., being equal chords of the circumscribed circle, are equidistant from the centre 0: hence, if a circle be described from 0 as a centre, with OP as a radius, it will be tangent to all of the sides of the polygon, and consequently, will be inscribed within it; which was to be proved.
Scholium. If the circumference of a circle be divided into equal arcs, the chords of these arcs will be sides of a regular inscribed polygon.
For, the sides are equal, because they are chords of equal arcs, and the angles are equal, because they are measured by halves of equal arcs.
If the vertices A, B, C, &c., of a regular inscribed polygon be joined with the centre O, the triangles thus formed will be equal, because their sides are equal, each to each hence, all of the angles about the point O are equal to each other.
1. The CENTRE OF A REGULAR POLYGON, is the common centre of the circumscribed and inscribed circles.
2. The ANGLE AT THE CENTRE, is the angle formed by drawing lines from the centre to the extremities of either side.
The angle at the centre is equal to four right angles divided by the number of sides of the polygon.
3. The APOTHEM, is the distance from the centre to either side.
The apothem is equal to the radius of the inscribed circle.