THEOREM XXV.

If from the right angle of a right-angled triangle, a per· pendicular is drawn to the hypotenuse ;

1. The perpendicular divides the triangle into two similar triangles, each of which is similar to the whole triangle.

2. The perpendicular is a mean proportional between the segments of the hypotenuse.

3. The segments of the hypotenuse are in proportion to the squares on the adjacent sides of the triangle.

4. The sum of the squares on the two sides is equivalent to the square on the hypotenuse.

Let BAC be a triangle, right angled at A; and draw AD perpendicular to BC.

1. The two A's, ABC and ABD, B

have the common angle, B, and the right angle BAC = the right angle BDA; therefore, the third 's are equal, and the two A's are similar by Th. 17, Cor. 1. In the same manner we prove the ▲ ADC similar to the A ABC; and the two triangles, ADB, ADC, being similar to the same ▲ ABC, are similar to each other.

2. As similar triangles have the sides about the equal angles proportional, (Def. 16), we have

or, the perpendicular is a mean proportional between the seyments of the hypotenuse.

Dividing Eq. (1) by Eq. (2), member by member, wo

which, in the form of a proportion, is

CA: BA':: CD: BD;

that is, the segments of the hypotenuse are proportional to the squares on the adjacent sides.

4. By the addition of (1) and (2), we have

BA2 + CA2 = BC (BD + CD) = BC;

that is, the sum of the squares on the sides about the right angle is equivalent to the square on the hypotenuse. This is another demonstration of 'Theorem 39, B. L.

Hence the theorem, if from the right angle of a right angled triangle, etc.

BOOK III.

OF THE CIRCLE, AND THE INVESTIGATION OF THEO REMS DEPENDENT ON ITS PROPERTIES.

DEFINITIONS.

1. * A Curved Line is one whose consecutive parts, however small, do not lie in the same direction.

2. A Circle is a plane figure bounded by one uniformly curved line, all of the points of which are at the same listance from a certain point within, called the center

3. The Circumference of a circle is the curved line that bounds it.

4. The Diameter of a circle is a line passing through the center, and terminating at both extremities in the circumference. Thus, in the figure, Cis the center of the circle, the curved line AGBD is the circumference, and AB is a diameter.

M

A

N

G

E

F

BL

H

5. The Radius of a circle is a line extending from the center to any point in the circumference. Thus, CD is a radius of the circle.

6. An Arc of a circle is any portion of the circumference.

* The first six of the above definitions have been before given among the general definitions of Geometry, but it was deemed advisable to reinsert them here.

7. A Chord of a circle is the line connecting the extremities of an arc.

8. A Segment of a circle is the portion of the circle on either side of a chord.

Thus, in the last figure, EGF is an arc, and EF is a chord of the circle, and the spaces bounded by the chord EF, and the two arcs EGF and EDF, into which it divides the circumference, are segments.

9. A Tangent to a circle is a line which, meeting the circumference at any point, will not cut it on being produced. The point in which the tangent meets the circumference is called the point of tangency.

10. A Secant to a circle is a line which meets the circumference in two-points, and lies a part within and a part without the circumference.

11. A Sector of a circle is a portion of the circle included between any two radii and their intercepted arc.

Thus, in the last figure, the line HL, which meets the circumference at the point D, but does not cut it, is a tangent, D being the point of tangency; and the line MN, which meets the circumference at the points P and Q, and lies a portion within and a portion without the circle, is a secant. The area bounded by the arc BD, and the two radii CB, CD, is a sector of the circle.

12. A Circumscribed Polygon is one all of whose sides are tangent to the circumference of the circle; and conversely, the circle is then said to be inscribed in the polygon. 13. An Inscribed Polygon is one the vertices of whose angles are all found in the circumference

B

F

E

of the circle; and conversely, the circle is then said to be circumscribed about the polygon.

14. A Regular Polygon is one which is both equiangu. lar and equilateral.

The last three definitions are illustrated by the last

figure.

THEOREM I.

Any radius perpendicular to a chord, bisects the chord, and also the arc of the chord.

Let AB be a chord, C the center of the circle, and CE a radius perpendicular to AB; then we are to prove that AD BD, and AE= EB.

=

Since C is the center of the circle, AC BC, CD is common to the two A's ACD and BCD, and the angles.

=

at D are right angles; therefore the two A's ADC and BDC are equal, and AD = DB, which proves the first part of the theorem.

Now, as ADDB, and DE is common to the two spaces, ADE and BDE, and the angles at D are right angles, if we conceive the sector CBE turned over and placed on CAE, CE retaining its position, the point B will fall on the point A, because AD = BD and AC = BC; then the arc BE will fall on the arc AE; otherwise there would be points in one or the other arc unequally distant from the center, which is impossible; therefore, the arc AE the arc EB, which proves the second part of the theorem.

Hence the theorem.

Cor. The center of the circle, the middle point of the chord AB, and of the subtended arc AEB, are three points in the same straight line perpendicular to the chord at its middle point. Now as but one perpendicular can be drawn to a line from a given point in that line, it follows:

1st. That the radius drawn to the middle point of any arc bisects, and is perpendicular to, the chord of the arc.