the equal distances of AB, CD from G (th. 17).* Then the two right-angled triangles, GAE, GCF, having the side GA equal the side GC (being radii), and the side GE equal the side GF, are identical (cor. 2, th. 26), and have the line AE equal to the line CF. But AB is the double of AE (th. 34), and CD is the double of CF; therefore AB is equal to CD (by ax. 6). Q. E. D.

Again, if the chord AB be equal to the chord CD, then will their distances from the center, GE, GF, also be equal to each other.

For, since in the right-angled triangles AEG, CFG, AE the half of AB is equal to CF, the half of CD, and the radii GA, GC are equal, therefore the third sides are equal (cor. 2, th. 26), or the distance GE is equal the distance GF. Q. E. D.

THEOREM XXXVI.

A line perpendicular to a radius at its extremity is a tangent to the circle.

Let the line ADB be perpendicular A DEB to the radius CD of a circle; then is any other point, except D, as E of the line AB, without the circle. For CE, an oblique line, is greater than the per

pendicular CD (th. 17), or greater than the radius. Hence, the line AB having but one point, D, in common with the circle, is a tangent (def. 56).

THEOREM XXXVII.

When a line is a tangent to a circle, a radius drawn to the point of contact is perpendicular to the tangent. For if oblique, a line shorter can be drawn perpendicular to the tangent, and the tangent must then pass within the circle, which is contrary to definition.

Corol. 1. Hence, conversely, a line drawn perpendicular to a tangent, at the point of contact, passes through the center of the circle; for there can be but one perpendicular to a given line through a given point.

By the distance of a point from a line is understood the shortest distance.

Corol. 2. If any number of circles touch each other at the same point, their centers must be in the same line perpendicular to their common tangent; for the perpendicular to the tangent at the common point must pass through the center of each.

THEOREM XXXVIII.

The angle formed by a tangent and chord is measured by half the arc of that chord.

Let AB be a tangent to a circle, and A CD a chord drawn from the point of contact C; then is the angle BCD measured by half the arc CFD, and the angle ACD measured by half the arc CGD.

C B

F

Draw the radius EC to the point of contact, and the radius EF perpendicular to the chord at H.

Then the radius EF, being perpendicular to the chord CD, bisects the arc CFD (th. 34). Therefore CF is half the arc CFD.

But the angle CEF is equal to the angle BCD, because the sides of the two angles are respectively perpendicular to each other, and consequently have the same difference of direction. Moreover, the angle CEF is measured by the arc CF (def. 10, note), which is the half of CFD; therefore the equal angle BCD must also have the same measure, namely, half the arc CFD of the chord CD.

Again, GEF being a diameter, CG is the supplement of CF, and is equal to GD, the supplement of FD. .. CG is half the arc CGD. Now, since the line CE, meeting FG, makes the sum of the two angles at E equal to two right angles (th. 6), and the line CD makes with AB the sum of the two angles at C equal to two right angles; if from these two equal sums there be taken away the parts or angles CEH and BCH, which have been proved equal, there remains the angle CEG, equal to the angle ACH. But the

former of these, CEG, being an angle at the center, is measured by the arc CG (def. 10, note); consequently the equal angle ACD must also have the same measure CG, which is half the arc CGD of the chord CD. Q. E. D.

Corol. 1. The sum of two right angles is measured by half the circumference. For the two angles BCD, ACD, which make up two right angles, are measured by the arcs CF, CG, which make up half the circumference, FG being a diameter.

Corol. 2. Hence, also, one right angle must have for its measure a quarter of the circumference, or 90 degrees.

THEOREM XXXIX.

An angle at the circumference of a circle is measured by half the arc that subtends it.

Let BAC be an angle of the circum- D ference; it has for its measure half the arc which subtends it.

B

A

E

For, suppose the tangent DE to pass through the point of contact A; then, the angle DAC being measured by half the arc ABC, and the angle DAB by half the arc AB (th. 38), it follows, by equal subtraction, that the difference, or angle BAC, must be measured by haif the arc BC, upon which it stands. Q. E. D.

Corol. 1. All angles in the same segment of a circle, or standing on the same arc, are equal to each other. Corol. 2. An angle at the center of a circle is double the angle at the circumference, when both stand on the same arc.

Corol. 3. An angle in a semicircle is a right angle.

THEOREM XL.

Any two parallel chords intercept equal arcs. Let the two chords AB, CD be parallel; then will the arcs AC, BD be equal; or AC = BD.

B

Draw the line BC. Then, because the A lines AB, CD are parallel, the alternate angles B and C are equal (th. 10). But the angle at the circumference B is measured by half the arc AC (th. 39); and the other equal angle at the circumference C is measured by half the arc BD; therefore the halves of the arcs AC, BD, and consequently the arcs themselves, are also equal. Q. E. D.

THEOREM XLI.

When a tangent and chord are parallel to each other, they intercept equal arcs.

Let the tangent ABC be parallel to A B the chord DF; then are the arcs BD, BF equal; that is, BD = BF.

D

C

F

But the

Draw the chord BD. Then, because the lines AB, DF are parallel, the alternate angles D and B are equal (th. 10). angle B, formed by a tangent and chord, is measured by half the arc BD (th. 38); and the other angle at the circumference D is measured by half the arc BF (th. 39); therefore the arcs BD, BF are equal. Q. E. D.

THEOREM XLII.

When two lines, meeting a circle each in two points, cut one another, either within or without it, the rectangle of the parts of the one is equal to the rectangle of the parts of the other, the parts of each being measured from the point of meeting to the two intersections with the circumference.

Let the two lines AB, CD meet each other in E; then the rectangle of AE, EB will be equal to the rectangle of CE, ED. Or, AE. EB = CE. ED.

For through the point E draw the diame

AFC

G

ter FG; also, from the center H draw the radius DH, and draw HI perpendicular to CD.

E CA

G

H

Then, since DEH is a triangle, and the perpendicular HI bisects the chord CD (th. D 34), the line CE is equal to the difference of the segments DI, EI, the sum of them being DE. Also, because H is the center of the circle, and the radii DH, FH, GH are all equal, the line EG is equal to the sum of the sides DH, HE; and EF is equal to their difference.

But the rectangle of the sum and difference of the two sides of a triangle is equal to the rectangle of the sum and difference of the segments of the base (cor., th. 27); therefore the rectangle of FE, EG is equal to the rectangle of CE, ED. In like manner, it is proved that the same rectangle of FE, EG is equal to the rectangle of AE, EB. Consequently, the rectangle of AE, EB is also equal to the rectangle of CE, ED (ax. 1). Q. E. D.

D

E

G

Corol. 1. When one of the lines in the second case, as DE, by revolving about the point E, comes into the position of the tangent EC or ED, the two points C and D running into one; then the rectangle of CE, ED becomes the square of CE, because CE and DE are then equal. Consequently, the rectangle of the parts of the secant, AE. EB, is equal to the square of the tangent, CE2.

Corol. 2. Hence both the tangents EC, EF, drawn from the same point E, are equal; since the square of each is equal to the same rectangle or quantity AE. EB.

THEOREM XLIII.

In equiangular triangles, the rectangles of the corresponding or like sides, taken alternately, are equal. Let ABC, DEF be two equiangular triangles, having the angle A the angle D, the angle B =

=