the BHM, BMH, CMR, and CRM are equal, $209 (being measured by halves of equal arcs); .. the ABHM and CMR are equal, § 107 (having a side and two adjacent ▲ of the one equal respectively to a side and two adjacent of the other). § 241 .. the polygon A B C D, etc., is equiangular. ). Ax. 6 .. the sides A B, BC, C D, etc. are equal, and the polygon A B C D, etc. is equilateral. Therefore the circumscribed polygon is regular and similar to the given inscribed polygon. $ 372 Q. E F. Ex. Let R denote the radius of a regular inscribed polygon, r the apothem, a one side, A one angle, and C the angle at the centre; show that 1. In a regular inscribed triangle a = R √3, 2. In an inscribed square a = R√√2, r = C = 90°. 3. In a regular inscribed hexagon a A 120°, C = 60°. = 4. In a regular inscribed decagon 1'= = r = R, R √2, A = 90°, R, r R V10+2 √5, A = 144°, C=36°, R (√5 PROPOSITION XXI. PROBLEM. 401. To find the value of the chord of one-half an arc, in terms of the chord of the whole arc and the radius of the circle. Let A B be the chord of arc AB and AD the chord of one-half the arc A B. It is required to find the value of A D in terms of A B and R (radius). From D draw D H through the centre 0, and draw O A. HD is to the chord A B at its middle point C, § 60 (two points, O and D, equally distant from the extremities, A and B, determine the position of a to the middle point of A B). 403. To compute the ratio of the circumference of a circle to its diameter, approximately. Let C be the circumference and R the radius of a It is required to find the numerical value of π. We make the following computations by the use of the formula obtained in the last proposition, 384 AD = = = Length of Side. Perimeter. 6.21165708 2 √4-(.51763809)2 .26105238 6.26525722 2-V4-(.26105238)2 .13080626 6.27870041 (.13080626)2 .06543817 6.28206396 2 √4-(.06543817)2 .03272346 6.28290510 V2 4-(.03272346)2 .01636228 6.28311544 768 AD=√2-√4-(.01636228)2 .00818121 6.28316941 Hence we may consider 6.28317 as approximately the circumference of a O whose radius is unity. ON ISOPERIMETRICAL POLYGONS. SUPPLEMENTARY. 404. DEF. Isoperimetrical figures are figures which have equal perimeters. 405. DEF. Among magnitudes of the same kind, that which is greatest is a Maximum, and that which is smallest is a Minimum. Thus the diameter of a circle is the maximum among all inscribed straight lines; and a perpendicular is the minimum among all straight lines drawn from a point to a given straight line. PROPOSITION XXIII. THEOREM. 406. Of all triangles having two sides respectively equal, that in which these sides include a right angle is the maxi Let the triangles ABC and EBC have the sides A B and BC equal respectively to EB and BC; and let the angle ABC be a right angle. The AABC and E B C, having the same base B C, are to each other as their altitudes A B and ED, (having the same base are to each other as their altitudes). (a is the shortest distance from a point to a straight line). $326 § 52 Hyp. Q. E. D. |