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1. Every point in the bisector of an angle is equally distant from the sides of the angle; and every point not in the bisector, but within the angle, is unequally distant from the sides of the angle.

2. To find the center of a given arc of a circle.

3. If three or more straight lines passing through a common point intersect two parallels, the corresponding segments of the parallels are proportional.

4. If through a fixed point without a circle a secant is drawn, the product of the whole secant and its external segment has the same value, in whatever direction the secant is drawn.

5. Of all isoperimetric polygons having the same number of sides, the regular polygon is the maximum.


1. Parallel lines intersecting a plane make equal angles with it.

2. A straight line makes equal angles with parallel planes.

3. A sphere may be inscribed in any given tetrahedron. 4. The volume of a pyramid is one-third the volume of a prism having the same base and altitude.

5. Assuming the earth to be a sphere, what portion of its surface is contained in the zone extending 30° on each side of the equator?



1. To find a point equidistant from two given points and also equidistant from two intersecting lines, the points being in the plane of the intersecting lines.

2. The three medial lines of a triangle intersect in a point which is two-thirds the distance from the vertex of each angle to the middle of the opposite side.

3. Construct the circumference of a circle which shall be tangent to two given parallels and pass through a given point between them.

4. The rectangle of two sides of a triangle is equal to the rectangle of the segments of the third side formed by the bisector of its opposite angle, plus the square of the bisector.

5. Inscribe an equilateral triangle in a given circle, and express the length of its side in terms of the radius of the circle.

6. Find the locus of a point in space equidistant from three given points not in the same straight line.

7. The altitude of a given cone is a, and the radius of its base is b: find (1) its volume, and (2) the altitude and radius of the base of a similar cone whose volume is n times as great.

8. To find the radius of a given material sphere with a rule and compasses.


1. Two straight lines which are parallel to a third straight line are parallel to each other.

2. Two sides of a triangle, and the angle opposite one of them, are given. Construct the triangle giving all possible solutions.

3. The perimeter of the circumscribed equilateral triangle is double that of the similar inscribed triangle.

4. Similar triangles are as the squares on any two homologous lines.

5. The circumference is the limit of the perimeters of the circumscribed and inscribed similar regular polygons when the number of sides is indefinitely increased.


6. The area of a certain polygon is S square feet. the area of the similar polygon whose perimeter is in the ratio of m to n to that of the given polygon.


1. Three parallel planes cut proportional segments from straight lines intersecting them.

2. A plane passed through two diagonally opposite edges of a parallelopiped divides it into two equivalent triangular prisms.

3. The sides of a right triangle are a and b. Compare the volumes of the solids of revolution generated about these sides as axes.


4. Define polar triangles. Between what limits must the sum of the angles of all spherical triangles lie? proof.

5. Find the volume of a sphere whose area is 167.

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