Ex. 6. A triangle, given the three medians.

[SUG. Reduce this to the preceding Ex. See Art. 187.]

Ex. 7. An isosceles trapezoid, given the bases and an angle.

Ex. 8. An isosceles trapezoid, given the bases and a diagonal. Ex. 9. A trapezoid, given the four sides.

Ex. 10. A trapezoid, given the bases and the two diagonals. [SUG. Reduce to Art. 283 by producing the lower base a distance equal to the upper base, etc.]

297. Construction of circles. The construction of a required circle is frequently a good illustration of the preceding method of reducing one construction problem to another. For the construction of a circle frequently reduces to the problem of finding a point (the center of the circle) which answers given conditions. (See Art. 294.)

Ex. Construct a circle which shall touch two given intersecting lines and have its center in another given line.

This problem is equivalent to the problem of finding a point which shall be in a given line and be equidistant from two other given lines. (See Ex. 3, p. 168.) In some cases, however, the construction of a required circle must be made by an independent method.

EXERCISES. GROUP 27

CONSTRUCTION OF CIRCLES

Construct a circle with given radius, 7,

Ex. 1. Which passes through a given point and touches a given line.

Ex. 2. Which has its center in a given line and touches another given line.

Ex. 3. Which passes through two given points.

Construct a circle

Ex. 4. Which touches two given parallel lines and passes through a given point.

Ex. 5. Which passes through two given points and has its center on a given line.

Ex. 6. Which touches three given lines, two of which are parallel.

Ex. 7. Which passes through a given point and touches a given line BC at a given point B.

[SUG. Draw AB and at B construct a to BC.]

Ex. 8. Which touches a given line and also touches a given circle at a given point A.

B

Ex. 9. Which touches a given line AB at a given point and touches a given circle.

EXERCISES. CROUP 28

PROBLEMS SOLVED BY VARIOUS METHODS

Ex. 1. Through a given point draw a line which shall cut two given intersecting lines so as to form an isosceles triangle.

Ex. 2. Construct an isosceles triangle, given the altitude and one leg.

Ex. 3. In a given circumference find a point equidistant from two given intersecting lines.

Ex. 4. Draw a circle which shall touch two given intersecting lines, one of them at a given point.

Ex. 5. Draw a line which shall be terminated by the sides of a given angle, shall equal a given line, and be parallel to another given line.

Ex. 6. Construct a triangle, given one side, an adjacent angle, and the difference of the other two sides.

Ex. 7. Find a point in a given circumference at a given distance from a given point.

Ex. 8. Construct a parallelogram, given a side, an angle, and a diagonal.

Ex. 9. Through a given point within an angle, draw a straight line terminated by the sides of the angle and bisected by the given point.

[SUG. Draw a line from the vertex of the angle to the given point and produce it its own length through the point.]

Ex. 10. Construct a triangle, given the vertex angle and the segments of the base made by the altitude.

[SUG. Use Art. 291.]

Ex. 11. Construct an isosceles triangle, given the angle at the vertex and the base.

Ex. 12. Draw a circle with given radius which shall touch a given circle at a given point.

Ex. 13. Construct a right triangle, given the hypotenuse and the altitude upon the hypotenuse.

Ex. 14. Construct a triangle, given the base and the altitudes upon the other two sides.

[SUG. Construct a semicircle on the given base as a diameter.]

Ex. 15. Find a point in one side of a triangle equidistant from the other two sides.

Ex. 16. Construct a triangle, given the altitude and the angles at the extremities of the base.

Ex. 17. Construct a rhombus, given an angle and a diagonal.

Ex. 18. Draw a circle which shall pass through two given points and have its center equidistant from two given parallel lines.

Ex. 19. Construct a triangle, given one side, an adjacent angle and the radius of the circumscribed circle.

Ex. 20. In a given circle draw a chord equal to a given line and parallel to another given line.

[SUG. Find the distance of the given chord from the center, by constructing a right triangle of which the hypotenuse and one leg are given.]

Ex. 21. Construct a triangle, given an angle, the bisector of that angle, and the altitude from another vertex.

Ex. 22. Find the locus of the points of contact of tangents drawn from a given point to a series of circles having a given center.

Ex. 24. Given a line AB and two points C and D on the same side

of AB; find a point P in AB such that CP+PD shall be a minimum.

Ex. 25. Draw a common external

tangent to two given circles.

Ex. 26. Draw a common internal

tangent to two given circles.

BOOK III

PROPORTION. SIMILAR POLYGONS

THEORY OF PROPORTION

298. Ratic has been defined, and its use briefly indicated in Arts. 245, 246.

299. A proportion is an expression of the equality of two or more equal ratios.

As,

or a : b = c :d.

This reads, "the ratio of a to b equals the ratio of c to d," or, "a is to b as c is to d."

300. The terms of a proportion are the four quantities. used in the proportion. In a proportion

the antecedents are the first and third terms;

the consequents are the second and fourth terms;

the extremes are the first and last terms;

the means are the second and third terms.

A fourth proportional is the last term of a proportion (provided the means are not equal).

Thus, in a bed, d is a fourth proportional.

:

301. A continued proportion is a proportion in which each consequent and the next antecedent are the same.

Thus, a:b=b:c=c : d=d: e is a continued proportion. A mean proportional is the middle term in a continued. proportion containing but two ratios.