1.

To divide a given line A B into two equal parts; and to erect a perpendicular through the middle.

With the end A and B as centers, draw the dotted circle arcs with a radius greater than half the line. Through the crossings of the arcs draw the perpendicular C D,which divides the line into two equal parts.

2.

From a given point C on the line A B, erect a perpendicular C D.

With C as a center, draw the dotted circle arcs at A and B equal distances from C. With A and B as centers, draw the dotted circle arcs at D. From the crossing D draw the required perpendicular D C.

3.

From a given point C at a distance from the line A B, draw a perpendicular to the line.

With C as a center, draw the dotted circle arc so that it cuts the line at A and B. With A and B as centers, draw the dotted cross arcs at D with equal radii. Draw the required perpendicular through C and crossing D.

4.

At the end of A to a given line A B, erect a perpendicular A C.

With the point D as a center at a distance from the line, and with A D as radius, draw the dotted circle arc so that it cuts the line at E through E and D, draw the diameter E C; then join C and A, which will be the required perpendicular.

* Copyright, 1895, by J. B. Lippincott Co. Published by special permission of, and arrangement with Messrs. J. B. Lippincott Co.

5.

Through a given point Cat a distance from the line A B, draw a line C D parallel to A B.

With C as a center, draw the dotted arc E D, with E as a center, draw through C the dotted arc F. C. With the radius F C and E as a center, draw the cross arc at D. Join C with the cross at D, which will be the required parallel line.

6.

On a given line A B and at the point B, construct an angle equal to the angle C D E.

With D as a center, draw the dotted arc C E; and with the same radius and B as a center, draw the arc G F; then make G F equal to C E; then join B F, which will form the required angle, F B G = C D E.

7.

Divide the angle Ac B into two equal parts.

With C as a center, draw the dotted arc D E; with D and E as centers, draw the cross arcs at F with equal radii. Join C F, which divides the angle into the required parts.

Angles A C F = F C B = ½(A C B).

8.

Divide an angle into two equal parts, when the lines do not extend to a meeting point.

Draw the lines C D and C E parallel, and at equal distances from the lines A B and F G. With C as a center, draw the dotted arc B G; and with B and G as centers, draw the cross arcs H. Join C H, which divides the angle into the required equal parts.

9.

To construct a parallelogram, with the given sides A and B and angle C.

Draw the base line D E, and make the angle F D E = C; lines D E = B and D F = A; complete the parallelogram by cross arcs at G, and the problem is thus solved.

10.

To divide the line A B in the same proportion of parts as A C.

Join C and B, and through the given divisions 1, 2, and 3 draw lines parallel with C B, which solves the problem.

11.

To find the center of a circle which will pass through three given points A, B, and C.

With B as a center, draw the arc D E F G; and with the same radius and A as a center, draw the cross arcs D and F; also with C as a center, draw the cross arcs E and G. Join D and F, and also E and G, and the crossing o is the required center of the circle.

12.

To construct a square upon a given line A B.

With A B as radius and A and B as centers, draw the circle arcsA E D and B E C. Divide the arc B E in two equal parts at F, and with E F as radius, and E as center, draw the circle C F D. Join A and C B and D, C and D, which completes the required square.

13.

Through a given point A in a circumference, draw a tangent to the circle.

Through a given point A and center C, draw the line B C. With A as a center, draw the circle arcs B and C; with B and C as centers, draw the cross arcs D and E; then join D and E, which is the required tangent.

14.

From a given point A outside of a circumference, draw a tangent to the circle.

Join A and C, and upon AC as a diameter draw the half circle Abc, which cuts the given circle at B. Join A and B, which is the required tangent.

15.

To draw a circle with a given radius R, that will tangent the circle A B C at C.

Through the given point C, draw the diameter A C extended beyond D; from C set off the given radius R to D; then D is the center of the required circle, which tangents the given circle at C.

16.

To draw a circle with a given radius R, that will tangent two given circles.

Join the centers A and B of the given circles Add the given radius R to each of the radii of the given circle, and draw the cross arcs C, which is the center of the circle required to tangent the other two.

17.

To draw a tangent to two circles of different diameters.

Join the centers C and c of the given circles, and extend the line to D; draw the radii A C and a c parallel with one another. Join A a, and extend the line to D. On C D as a diameter, draw the half circle C e D; on c D as a diameter, draw the half circle c f D; then the crossings e and f are the tangenting points of the circles.

18.

To draw a tangent between two circles.

Join the centers C and c of the given circles; draw the dotted circle arcs, and join the crossing m, n, which line cuts the center line at a. With a C as a diameter, draw the half circle a f C; and with a c as a diameter, draw the half circle c e a; then the crossings e and f are the tangenting points of the circles.

19.

With a given radius r, draw a circle that will tangent the given line A B and the given circle C D.

Add the given radius r to the radius R of the circle, and draw the arc c d. Draw the line c e parallel with and at a distance r from the line A B. Then the crossing c is the center of the required circle that will tangent the given line and circle.