226. Find the locus of the centers of circles which touch a given circle at a given point.

227. Two equal given circles touch each other, and each touches one side of a right angle; find the locus of their point of contact.

228. In a given line find a point at which a given sect subtends a given angle.

229. Describe a circle of given radius to touch a given line and have its center on another given line.

230. At any point in the circle circumscribing a square, show that one of the sides subtends an angle thrice the others.

231. Divide a given arc of a circle into two parts which have their chords in a given ratio.

232. The sect of a common tangent between its points of contact is a mean proportional between the diameters of two tangent circles.

233. Any regular polygon inscribed in a circle is a mean proportional between the inscribed and circumscribed regular polygons of half the number of sides.

234. The circumcenter, the centroid, the mediocenter, and the orthocenter form a harmonic range,



The most instructive problems in geometry of three dimensions are made by generalizing those first solved for plane geometry. This way of getting a theorem in solid geometry is often difficult, but a number of the exercises here given are specially adapted for it.

In the author's “Mensuration" (published by Ginn & Co.) are given one hundred and six examples in metrical geometry worked out completely, and five hundred and twenty-four exercises and problems, of which also more than twenty are solved completely, and many others have hints appended.

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