Plane problems in elementary geometry; or, Problems on the elementary conic sections the point, straight line, and circle - Softcover
Synopsis
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1899 Excerpt: ...so as to form two, at least, of the three interior angles included among them. 2d. Bisect these two angles, and the intersection of the two bisecting lines will be the centre of the required circle. Principle.--Tangents to a circle, from the same point, make equal angles with the diameter, produced, through that point. Remark.--By producing each line beyond its intersection with the other two, and by then bisecting the external angles, three circles may be found, each tangent to the three given lines and external to the triangle inclosed by them. If, however, two of the lines are parallel, only two tangent circles, in all, can be drawn; as will appear on bisecting the angles made by the parallels with the third, or secant line. Prop.. 84. To draw a Circle, tangent to two given straight lines, and to a given circle. Let AB and DF be the given straight lines, and the circle, with radius Co, the given circle. (Fig. 63.) 1st. By Prob. 31, or 48, b, construct the bisecting line, QC, of tho angle included by the given lines. QC is perpendicular to tho chord, on, of the points of contact, o and n. 2d. At t draw the tangent ts, which is perpendicular to CQ. 3d. With s as a centre, and a radius st, describe an arc at r, making sr equal to st. 4th. Make rQ, perpendicular to AB. 5th. With Q as a centre, and Qr as a radius, describe the circle rte, which will be the required tangent circle. Principle.--In a quadrilateral--tsrQ--if two sides--ts and sr--and the angles between these and the remaining sides--stQ, and «rQ--are equal, those remaining sides--tQ and rQ--will be equal. Remarks.--(a) Another circle, tangent to either of those now drawn, could be drawn by the above solution. Hence we have the general problem: To draw any number of circles tangent to each ot...
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