of the square.

To do this, it will be clear to you that the radius of the circle will be the half diagonal O C, so that the four angles of the square will be four points in the circle. Now produce the lines F G and H I beyond the square, and mark on them the length OD; this will give points N, P, Q, R and you will then be able to draw your circle through the points C, Q, A, P, B, R, D, N.

We now come to the third lesson to be obtained from this very useful figure, namely

To inscribe a Square in a Circle.

We will suppose the circle to be drawn, and it would then be called "the given circle."

Draw the diameter Q R, and bisect it by the diameter N P.

Bisect the four right angles at the centre by the diameters C B and D A.

Then draw the lines C D, A B, C A, and D B, which will complete the square in the circle.

is

The fourth lesson to be derived from this figure,

To describe a Square about a given Circle. For this purpose we will use the smaller circle.

Draw the diameter H I, and then F G at right angles to it.

Now draw a line at F, and another at G, parallel to H I-namely, lines C D and A B.

Next draw at H and I the lines C A and D B. These will meet the former two lines in the points C, D, A, B, and will thus complete the square about the circle.

TANGENTS.

A Straight line which touches a Circle at one point, is called a Tangent.

A tangent is always at right angles to one radius.

The line A B in the last figure is a tangent, and you will see that it touches the circle at only one point, and is at right angles to the radius G O. The three other sides of the square are also tangents, and are at right angles to the radii drawn from the points at which they touch.

43

E

Here we have another figure (43) from which we shall be able to obtain several lessons, the first of which is

To inscribe a Circle in an Equilateral Triangle.

Of course you must be very careful that your triangle is really equilateral. Find the middle of each of the sides-namely, D, E, F and draw lines from these points into the angles of the triangle.

The lines in an equilateral triangle will bisect not only the sides, but the angles as well, and the point O will be the centre of the figure. Now, it will be clear that O D will be the radius of the required circle, and as O E and O F are equal to O D, you have three radii of the circle which touch the triangle.

But these lines pass straight through the centre into the opposite angles of the triangle, and can be turned into diameters, instead of radii, by setting off on them the length of the radius from O on O C, O A, and O B. You will thus obtain three more points in the circle, which you can now draw through G, D, H, F, I, E.

And now we will use the figure for a second lesson, namely

To describe an Equilateral Triangle about a given Circle.

Divide the circle into six equal parts. (The sixth part of a circle is equal to its radius.)

Draw the six radii, D, E, F, G, H, I, and produce the three O G, O H, and O I.

Now, at points D, E, and F, draw lines at right angles to the radii D O, E O, and F O; these, you will remember, are called tangents, and will meet the

three produced radii, G, H, I, in the points A, B, C, thus completing the equilateral triangle described about the circle.

But we can use the figure for a third lesson; that is

To inscribe an Equilateral Triangle in a Circle.

Draw the diameter C J, and on each side of J, mark off on the circle the length of the radius; viz., points A and B.

Draw a line from C to A, another from C to B, and another from A to B, which will complete the equilateral triangle inscribed in the circle.

From this figure we can also learn how

To describe a Circle about an Equilateral Triangle.

Having bisected the sides of the triangle, draw the lines B D, A E, and C F, and carry them on beyond the triangle. Mark on these three lines, from O, the length of the radius O C, by which you will obtain the points J, K, L. Draw your circle

through points C, K, A, J, B, L, and you will thus complete your figure as desired.