PROBLEM VI. In a right-angled triangle, having given the base and the sum of the perpendicular and hypotenuse, to find these two sides.

PROB. VII.-Given, the base and altitude of a triangle, to divide it into three equal parts, by lines parallel to the base.

PROB. VIII.—In any equilateral A, given the length of the three perpendiculars drawn from any point within, to the three sides, to determine the sides.

PROB. IX.—In a right-angled triangle, having given the base, (3), and the difference between the hypotenuse and perpendicular, (1), to find both these two sides.

PROB. X.-In a right-angled triangle, having given the hypotenuse, (5), and the difference between the base and perpendicular, (1), to determine both these two sides.

PROB. XI.-Having given the area of a rectangle inscribed in a given triangle, to determine the sides of the rectangle.

PROB. XII.—In a triangle, having given the ratio of the two sides, together with both the segments of the base, made by a perpendicular from the vertical angle, to determine the sides of the triangle.

PROB. XIII.—In a triangle, having given the base, the sum of the other two sides, and the length of a line drawn from the vertical angle to the middle of the base, to find the sides of the triangle.

PROB. XIV.-To determine a right-angled triangle, having given the lengths of two lines drawn from the acute angles to the middle of the opposite sides.

PROB. XV.-To determine a right-angled triangle, having given the perimeter, and the radius of the inscribed circle.

PROB. XVI.-To determine a triangle, having given the base, the perpendicular, and the ratio of the two sides.

PROB. XVII.-To determine a right-angled triangle, having given the hypotenuse, and the side of the inscribed square.

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PROB. XVIII. To determine the radii of three equal circles inscribed in a given circle, and tangent to each other, and also to the circumference of the given circle.

PROB. XIX. —In a right-angled triangle, having given the perimeter, or sum of all the sides, and the perpendicular let fall from the right angle on the hypotenuse, to determine the triangle; that is, its sides.

PROB. XX.-To determine a right-angled triangle, having given the hypotenuse, and the difference of two lines drawn from the two acute angles to the center of the inscribed circle.

PROB. XXI. To determine a triangle, having given the base, the perpendicular, and the difference of the two other sides.

PROB. XXII. To determine a triangle, having given the base, the perpendicular, and the rectangle, or product of the two sides.

PROB. XXIII.-To determine a triangle, having given the lengths of three lines drawn from the three angles to the middle of the opposite sides.

PROB. XXIV. — In a triangle, having given all the three sides, to find the radius of the inscribed circle.

PROB. XXV. To determine a right-angled triangle, having given the side of the inscribed square, and the radius of the inscribed circle.

PROB. XXVI. To determine a triangle, and the radius of the inscribed circle, having given the lengths of three lines drawn from the three angles to the center of that circle.

PROB XXVII. To determine a right-angled triangle, having given the hypotenuse, and the radius of the inscribed ircle.

PROB. XXVIII.-The lengths of two parallel chords on the same side of the center being given, and their distance apart, to determine the radius of the circle.

PROD. XXIX. The lengths of two chords in the same

circle being given, and also the difference of their distances from the center, to find the radius of the circle.

PROB. XXX.-The radius of a circle being given, and also the rectangle of the segments of a chord, to determine the distance of the point at which the chord is divided, from the

center.

PROB. XXXI.-If each of the two equal sides of an isosceles triangle be represented by a, and the base by 2b, what will be the value of the radius of the inscribed circle?

PROB. XXXII. —From a point without a circle whose diameter is d, a line equal to d is drawn, terminating in the concave arc, and this line is bisected at the first point in which it meets the circumference. What is the distance of the point without from the center of the circle?

It is not deemed necessary to multiply problems in the application of algebra to geometry. The preceding will be a sufficient exercise to give the student a clear conception of the nature of such problems, and will serve as a guide for the solution of others that may be proposed to him, or that may be invented by his own ingenuity.

MISCELLANEOUS PROPOSITIONS.

We shall conclude this book, and the subject of Gconetry, by offering the following propositions, some theorems, others problems, and some a combination of both, -not only for the purpose of impressing, by application, the geometrical principles which have now been estab lished, but for the not less important purpose of culti vating the power of independent investigation.

After one or two propositions in which the beginner will be assisted in the analysis and construction, we shall leave him to his own resources, with the caution that a

patient consideration of all the conditions in each case, and not mere trial operation, is the only process by which he can hope to reach the desired result.

1. From two given points, to draw two equal straight lines, which shall meet in the same point in a given straight line.

Let A and B be the given points, and CD the given straight line. Produce the perpendicular to the straight line AB at its middle point, until it meets CD in G. It is then easily proved that G is the point in CD in which the equal lines from A and B must meet. That is, that AG BG.

A

E

B

C

G

2. From two given points on the same side of a given straight line, to draw two straight lines which shall meet in the given line, and make equal angles with it.

Let CD be the given line, and A and B the given points.

From B draw BE perpendicular to CD, and produce the perpendicular to F, making EF equal to BE; then draw AF, and from the point G, in which it intersects. CD, draw GB. Now, LBGE= LEGF=AGC. Hence, the angles BGD and AGC are equal, and the lines AG and BG meet

A

B

ED

F

in a common point in the line CD, and made equal angles with that line.

3. If, from a point without a circle, two straight lines be drawn to the concave part of the circumference, making equal angles with the line joining the same point and the center, the parts of these lines which are intercepted within the circle, are equal.

4. If a circle be described on the radius of another circle, any straight line drawn from the point where they meet, to the outer circumference, is bisected by the interior one.

5. From two given points on the same side of a line given in position, to draw two straight lines which shall contain a given angle, and be terminated in that line.

6. If, from any point without a circle, lines be drawn touching the circle, the angle contained by the tangents is double the angle contained by the line joining the points of contact and the diameter drawn through one of them.

7. If, from any two points in the circumference of a circle, there be drawn two straight lines to a point in a tangent to that circle, they will make the greatest angle when drawn to the point of contact.

8. From a given point within a given circle, to draw a straight line which shall make, with the circumference, an angle, less than any angle made by any other line drawn from that point.

9. If two circles cut each other, the greatest line that can be drawn through either point of intersection, is that which is parallel to the line joining their centers.

10. If, from any point within an equilateral triangle, perpendiculars be drawn to the sides, their sum is equal to a perpendicular drawn from any of the angles to the opposite side.

11. If the points of bisection of the sides of a given triangle be joined, the triangle so formed will be one fourth of the given triangle.

12. The difference of the angles at the base of any triangle, is double the angle contained by a line drawn from the vertex perpendicular to the base, and another bisecting the angle at the vertex.