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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,

EXERCISES IN GEOMETRY OF THREE DIMENSIONS AND

IN MENSURATION.

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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