By William J. M'Clelland

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**Read or Download A treatise on the geometry of the circle and some extensions to conic sections by the method of reciprocation, with numerous examples. PDF**

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**Additional info for A treatise on the geometry of the circle and some extensions to conic sections by the method of reciprocation, with numerous examples.**

**Example text**

R, by the sides of the given one at the point. 1 distances, #, y, z, from the sides A A Now A = &#, 2 and is and consequently a since AI From 4- (2) A2 + A3 it is This result (a 2 A = A 3 = ^, - minimum when const. * .................................... ] This point is termed the Symmedian Point of the triangle, obvious from (3) that the lines joining it to the vertices of the given triangle make the same angles with the sides as the NOTE. as it is corresponding medians ; also since x^y^z_ax+by+cz_ ~ a~b~c~~ 2aA 6A 2 A 2cA MAXIMUM AND MINIMUM.

In the figure are similar, or all of the same will be observed that if the fifth circle The contacts kind, but it touches any two with contacts of opposite species, their transverse common tangents must be substituted in (1). We let 12' denote the transverse tangent to Ov r^ and 2, r2 ; then _ 12' 2 = 2 <5 12 -(r1 + r2) 2 . For example, if the circle O v r x is external and the remaining circles internal to 0, R the relation is written 23 n'+Sr M+iT 5^=o, with analogous expressions for all other cases.

Again in the triangle BOG 00 a maximum, and is is Where when these circles meet EG is a maximum therefore a diameter of the Then OPC is a right angle. Hence the triangle of given species escribed to a given one is that whose sides are perpendicular to OP, OQ, OR. circle OPQO. maximum X, COR. If the sides of the given escribed triangle be and a, /3, y the distances of from P, Q, R, yu, and i/, Xa + Hence required / u/3 + i>y = a minimum. find a point, given multiples of whose distances from three fixed points is a minimum when any to two of the multiples are together greater than the third.