Chapter 5

Euclid’s Elements Book III 202 Euclid’s Elements Book III

5.1 The center of a circle

The basic elements of a circle are defined in I.15-18. Definition (IV.7). A straight line is said to be fitted into a circle when its extremi- ties are on the circumference of the circle. We shall instead call this a chord of the circle. Euclid (III.1). To find the center of a given circle. Euclid takes two points A and B on the circle and constructs the perpendicular bisector of the segment AB to intersect the circle at C and E. The midpoint F of CE is the center of the circle. C

F G

A D B

E Given: A circle. To construct: The center of the circle. Construction: (1) Take a chord AB of the circle. (2) Construct the midpoint D of AB. [I.10] (3) Construct the perpendicular to AB at D [I.11] to intersect the circle C and E. (4) Construct the midpoint F of CE. F is the center of the circle. Proof by contradiction. Suppose it is not. Let G be the center. G is not on the line CE, for otherwise it is the midpoint F . We may assume ∠BDG < ∠BDF. In triangles ADG and BDG, AD = BD,[D is midpoint of AB] AG = BG,[G is center of circle] DG = DG. ∴ triangleADG ≡ BDG, [SSS] and ∠BDG = ∠ADG =1rt.∠. But the greater angle ∠BDF is also a right angle. This is a contradiction. Therefore, F is the center of the circle. Q.E.D. 5.1 The center of a circle 203

Euclid (III.2). If on the circumference of a circle two points are taken at random, then the straight line joining the points will fall within the circle.

D

A F B

E Given: A chord AB of a circle, center D. To prove: Every point in the segment AB is inside the circle. Construction: Join DA, DB. Proof : Let E be a point in the segment AB. Join DE. Consider the side AE of triangle DAE extended to B. ∠DEB > ∠DAE. [I.16] But ∠DAE = ∠DBE. [I.5] Therefore, in triangle DEB, ∠DEB > ∠DBE, and DB > DE. [I.19] E is inside the circle since DE is less than the radius of the circle. Q.E.D.

Euclid (III.3). If in a circle a straight line through the center bisects a straight line not through the center, it also cuts it at right angles; and if it cut it at right angles, then it also bisects it.

C

E

F

D 204 Euclid’s Elements Book III

Euclid (III.4). If in a circle two straight lines cut one another which are not through the center, they do not bisect one another.

D F

A E C

B Given: Two chords AC and BD of a circle not containing the center F but inter- secting at E. To prove: E is not the common midpoint of AC and BD. Proof by contradiction: Suppose E is the midpoint of both AC and BD. Note that E and F are distinct points. ∠FEB is a right angle; [III.3] ∠FED is also a right angle; [III.3] One of these angles is part of the other. This contradicts Common Notion 5. Q.E.D.

5.2 Tangency of circles

Definitions. (III.2). A straight line is said to touch a circle which, meeting the circle and being produced, does not cut the circle. (III.3). Circles are said to touch one another which, meeting one another, do not cut one another.

Euclid (III.5). If two circles cut one another, they will not have the same center. (III.6). If two circles touch one another, then they do not have the same center.

C A C D

B E F

E F D B G

A 5.2 Tangency of circles 205

For let the two circles ABC, CDE touch one another at the point C; I say that they will not have the same center. For, if possible, let it be F ; let FC be joined, and let FEB be drawn through at random. Then, since the point F is the center of the circle ABC, FC is equal to FB. Again, since the point F is the center of the circle CDE, FC is equal to FE. But FC was proved equal to FE; therefore FE is also equal to FB, the less to the greater: which is impossible. Therefore F is not the center of the circles ABC, CDE. Therefore etc. Q.E.D.

Euclid (III.7). If on the diameter of a circle a point is taken which is not the center of the circle, and from the point straight lines fall upon the circle, then that is greatest on which passes through the center, the remainder of the same diameter is least, and of the rest the nearer to the straight line through the center is always greater than the more remote; and only two equal straight lines fall from the point on the circle, one on each side of the least straight line. (III.8). If a point is taken outside a circle and from the point straight lines are drawn through to the circle, one of which is through the center and the others are drawn at random, then, of the straight lines which fall on the concave circumference, that through the center is greatest, while of the rest the nearer to that through the center is always greater than the more remote, but, of the straight lines falling on the convex circumference, that between the point and the diameter is least, while of the rest the nearer to the least is always less than the more remote; and only two equal straight lines fall on the circle from the point, one on each side of the least.

Euclid (III.9). If a point be taken within a circle, and more than two equal straight lines fall from the point on the circle, the point taken is the center of the circle. 1 (III.10). A circle does not cut a circle at more points than two.

Euclid (III.11). If two circles touch one another internally, and their centers be taken, the straight line joining their centers, if it be also produced, will fall on the point of contact of the circles.

For let the two circles ABC, ADE touch one another internally at the point A, and let the center F of the circle ABC, and H the center G of ADE, be taken; I say that the straight line joined from G to F and produced will fall on A. For suppose it does not, but, if possible, let it fall as FGH, and let AF , AG be joined. Then, since AG, GF are greater than FA, that is, than FH, let FG be subtracted from each; therefore the remainder AG is greater than the remainder GR. But AG is equal to GD; therefore GD is also greater than GH, the less than the greater:

1If OA = OB = OC for three distinct points A, B, C on a circle, then O is the center of the circle. 206 Euclid’s Elements Book III

H

D A

G

F B

E

C which is impossible. Therefore the straight line joined from F to G will not fall outside; therefore it will fall at A on the point of contact. Therefore etc. Q.E.D. Euclid (III.12). If two circles touch one another externally, the straight line joining their centers will pass through the point of contact. III.13. A circle does not touch a circle at more points than one, whether it touch it internally or externally.

Euclid (III.14). In a circle equal straight lines are equally distant from the center, and those which are equally distant from the center are equal to one another.

Euclid (III.15). Of straight lines in a circle the diameter is greatest, and of the rest the nearer to the center is always greater than the more remote.

5.3 Tangents to a circle

Euclid (III.16). The straight line drawn at right angles to the diameter of a circle from its extremity will fall outside the circle, and into the space between the straight line and the circumference another straight line cannot be interposed; further the angle of the semicircle is greater, and the remaining angle less, than any acute rectilineal angle. Porism. From this it is manifest that the straight line drawn at right angles to the diameter of a circle from its extremity touches the circle. Euclid (III.17). From a given point to draw a straight line touching a given circle.

Let A be the given point, and BCD the given circle; thus it is required to draw from the point A a straight line touching the circle BCD. For let the center E of the circle be taken; [III.1] let AE be joined, and with center E and distance EA let the circle AF G be described; from D let DF be drawn at right angles to EA, 5.3 Tangents to a circle 207

A

D F

B

E

and let EF, AB be joined; I say that AB has been drawn from the point A touching the circle BCD.

Euclid (III.18). If a straight line touch a circle, and a straight line is joined from the center to the point of contact, the straight line so joined will be perpendicular to the tangent.

A

F F

C G C

(III.19). If a straight line touch a circle, and from the point of contact a straight line be drawn at right angles to the tangent, the center of the circle will be on the straight line so drawn.

Definitions. (III.4). In a circle straight lines are said to be equally distant from the center when the perpendiculars drawn to them from the center are equal. (III.5). And that straight line is said to be at a greater distance on which the greater perpendicular falls.

Definitions. (III.6). A segment of a circle is the figure contained by a straight line and a circum- ference of a circle. (III.7). An angle of a segment is that contained by a straight line and a circumfer- ence of a circle. 208 Euclid’s Elements Book III

(III.8). An angle in a segment is the angle which, when a point is taken on the circumference of the segment and straight lines are joined from it to the ends of the straight line which is the base of the segment, is contained by the straight lines so joined.

Definitions. (III.9). And, when the straight lines containing the angle cut off a circumference, the angle is said to stand upon that circumference. (III.10.) A sector of a circle is the figure which, when an angle is constructed at the center of the circle, is contained by the straight lines containing the angle and the circumference cut off by them. (III.11). Similar segments of circles are those which admit equal angles, or in which the angles equal one another.

5.4 Angle properties

Euclid (III.20). In a circle the angle at the center is double the angle at the circum- ference when the angles have the same circumference as base. (III.21). In a circle the angles in the same segment equal one another.

D A A B

A

C E F E

B B D C

G D

(III.22). The opposite angles of quadrilaterals in circles are equal to two right an- gles.

Euclid (III.23). On the same straight line there cannot be constructed two similar and unequal segments of circles on the same side. (III.24). Similar segments of circles on equal straight lines equal one another.

Euclid (III.25). Given a segment of a circle, to describe the complete circle of which it is a segment. 5.5 Constructions 209

Euclid (III.26). In equal circles equal angles stand on equal circumferences whether they stand at the centers or at the circumferences.

A

D G H

B C E F

(III.27). In equal circles angles standing on equal circumferences are equal to one another, whether they stand at the centers or at the circumferences. (III.28). In equal circles equal straight lines cut off equal circumferences, the greater equal to the greater and the less to the less. (III.29). In equal circles equal circumferences are subtended by equal straight lines. Euclid (III.30). To bisect a given circumference.

5.5 Constructions

Euclid (III.31). In a circle the angle in the semicircle is right, that in a greater segment less than a right angle, and that in a less segment greater than a right angle; further the angle of the greater segment is greater than a right angle, and the angle of the less segment is less than a right angle. A

C

F B (III.32). If a straight line touches a circle, and from the point of contact there is drawn across, in the circle, a straight line cutting the circle, then the angles which it makes with the tangent equal the angles in the alternate segments of the circle. Euclid (III.33). On a given straight line to describe a segment of a circle admitting an angle equal to a given rectilinear angle. 210 Euclid’s Elements Book III

(III.34). From a given circle to cut off a segment admitting an angle equal to a given rectilinear angle.

5.6 Intersecting chords theorems

Euclid (III.35). If in a circle two straight lines cut one another, the rectangle con- tained by the segments of the one is equal to the rectangle contained by the segments of the other.

A A D

F B D E G H

C E

B C

Euclid (III.36). If a point be taken outside a circle and from it there fall on the circle two straight lines, and if one of them cuts the circle and the other touches it, the rectangle contained by the whole of the straight line which cuts the circle and the straight line intercepted on it outside between the point and the convex circumference will be equal to the square on the tangent.

A

E B E

A F C C D B D 5.7 Summary of Book III 211

Euclid (III.37). If a point be taken outside a circle and from it there fall on the circle two straight lines, if one of them cuts the circle and the other falls on it, and if further the rectangle contained by the whole of the straight line which cuts the circle and the straight line intercepted on it outside between the point and the convex circumference is equal to the square on the straight line which falls on the circle, the straight line which falls on it will touch the circle.

E D

C

F

B A

5.7 Summary of Book III

Definition (III.1). Equal circles are those whose diameters are equal, or whose radii are equal.

• III.1: construction of the center of a circle.

• III.2: a circle is convex.

• III.3: A line through the center of a circle bisects a chord if and only if it is perpendicular to the chord.

• III.4: If two chords of a circle bisect each other, their common midpoint is the center of the circle.

• III.5,6: Two intersecting (or tangent) circles cannot be concentric.

• III.7,8: greatest and least lines through a given point and a point on a circle.

• III.9: An interior point of a circle is the center if more than two equal chords can be drawn through the point.

• III.10: Two circles intersect at at most two points.

• III.11,12: the line joining the centers of two tangent circles contains the point of tangency.

• III.13: Two circles may touch each other at at most one point. 212 Euclid’s Elements Book III

• III.14: Equal chords of a circle are equidistant from center.

• III.15: diameter of a circle is the greatest chord; longer chords of a circle are more remote from center.

• III.16: line perpendicular to diameter at extremity is external to circle.

• III.17: construction of tangent to a circle.

• III.18-19: tangent and perpendicular radius.

• III.20: angle at center is twice angle on circumference.

• III.21: equal segments contain equal angles.

• III.22: opposite angles of a cyclic quadrilateral are supplementary.

• III.23,24: Similar segments of circles on equal straight lines are equal to each other.

• III.25: construction of circle from a segment.

• III.26,27: In equal circles, equal arcs subtend equal angles on the circumfer- ence, and conversely.

• III.28,29: In equal circles, equal chords cut out equal segments, and con- versely.

• III.30: bisecting a given arc of a circle.

• III.31: angle in semicircle, major, and minor segments.

• III.32: angle between tangent and chord equal to angle in alternate segment.

• III.33,34: construction of segment of a circle with a given angle.

• III.35-37: Intersecting chords theorem. Chapter 6

Euclid’s Elements, Book IV

A 214 Euclid’s Elements, Book IV

6.1 Definitions

Definitions. (IV.1). A rectilineal figure is said to be inscribed in a rectilineal figure when the respective angles of the inscribed figure lie on the respective sides of that in which it is inscribed. (IV.2). Similarly a figure is said to be circumscribed about a figure when the respective sides of the circumscribed figure pass through the respective angles of that about which it is circumscribed.

Definitions. (IV.3). A rectilineal figure is said to be inscribed in a circle when each angle of the inscribed figure lies on the circumference of the circle. (IV.4). A rectilineal figure is said to be circumscribed about a circle when each side of the circumscribed figure touches the circumference of the circle.

Definitions. (IV.5). Similarly a circle is said to be inscribed in a figure when the circumference of the circle touches each side of the figure in which it is inscribed. (IV.6). A circle is said to be circumscribed about a figure when the circumference of the circle passes through each angle of the figure about which it is circumscribed.

Definition (IV.7). A straight line is said to be fitted into a circle when its extremi- ties are on the circumference of the circle.

6.2 Some basic constructions

Euclid (IV.1). Into a given circle to fit a straight line equal to a given straight line which is not greater than the diameter of the circle.

Euclid (IV.2). In a given circle to inscribe a triangle equiangular with a given triangle.

B E

F

C

D G

A H 6.3 Construction of square and regular hexagon 215

Euclid (IV.3). About a given circle to circumscribe a triangle equiangular with a given triangle. (IV.4). In a given triangle to inscribe a circle.

A

G

E I

B F C

Euclid (IV.5). About a given triangle to circumscribe a circle.

A A A D E

D E B C

D E

B C I I I

B C

Corollary. When the center of the circle falls within the triangle, the triangle is acute-angled; when the center falls on a side, the triangle is right-angled; and when the center of the circle falls outside the triangle, the triangle is obtuse-angled.

6.3 Construction of square and regular hexagon

Euclid (IV.6). In a given circle to inscribe a square. (IV.7). About a given circle to circumscribe a square.

Construct two perpendicular diameters. The endpoints of the diameters are the vertices of a square inscribed in the circle. (IV.6). The perpendiculars at these endpoints to the respective diameters bound a square which circumscribes the circle (IV.7). 216 Euclid’s Elements, Book IV

A G A F

E I B D B D

C K C H

Euclid (IV.8). In a given square to inscribe a circle. (IV.9). In a given square to circumscribe a circle. Euclid (IV.15). In a given circle to inscribe an equilateral and equiangular hexagon.

D

C E

G

B F

A

6.4 Construction of regular pentagon

Euclid (IV.10). To construct an isosceles triangle having each of the angles at the base double of the remaining one.

B

C

D A

E

Given: A segment AB. To construct: Triangle ABD such that ∠B = ∠D =2∠A. 6.4 Construction of regular pentagon 217

Construction: Construct (by II.11) a point C on AB such that the square on AC is equal to the rectangle contained by AB and CB, a point D such that AD = AB and CD = CA. To prove: In triangle ABD, ∠B = ∠D =2∠A. Proof : Since AB = AD, ∠B = ∠D. The rectangle contained by BA and BC is equal to the square on BD (by construc- tion). Therefore, BD is tangent to the circle ACD at D. (III.37) From this, ∠CAD = ∠CDB. (III.32) But also since CA = CD, ∠CAD = ∠ADC. Therefore, ∠D = ∠ADC + ∠CDB =2∠A.

Euclid (IV.11). In a given circle to inscribe an equilateral and equiangular pen- tagon. (IV.12). About a given circle to circumscribe an equilateral and equiangular pen- tagon. (IV.13). In a given pentagon, which is equilateral and equiangular, to inscribe a circle. (IV.14). About a given pentagon, which is equilateral and equiangular, to circum- scribe a circle.

A

F

B E

G H

C D

Euclid (IV.16). In a given circle to inscribe a fifteen-angled figure which shall be both equilateral and equiangular.

Corollary to IV.16. And, in like manner as in the case of the pentagon, if through the points of division on the circle we draw tangents to the circle, there will be cir- cumscribed about the circle a fifteen-angled figure which is equilateral and equian- gular. 218 Euclid’s Elements, Book IV

A

B

E

C D

6.5 Appendix: Gauss’s construction of a regular 17- gon

Circle (O) with perpendicular diameters PQand RS. 1 (1) A on OR with OA = 4 OP. (2) Construct the bisectors OB and OC of angle OAP. (3) D = OP ∩ C(C, A) and E = OP ∩ C(B,A).

R

A

Q P C O D B E (4) M = midpoint M of QD. (5) F = OS ∩ C(M,Q). (6) G on semicircle on OE, with OG = OF. (7) H = OP ∩ C(E,G). (8) P1H ⊥ OP. 6.6 Construction exercises for Books III and IV 219

R

P1 A

Q P M C O D B E H

F G

S 6.6 Construction exercises for Books III and IV

1. Through an intersection of two given circles, construct the longest line termi- nated by the two circles. 2. Construct a circle tangent to a given circle, with center on a given straight line, and pass through a given point on the given straight line. 3. Construct the tangents to a given circle perpendicular to a given straight line. 4. Construct a circle tangent to a given line at a given point, such that the tan- gents drawn to it from two given points on the given line are parallel. 5. Construct a triangle, given a base, the vertical angle, and the altitude. 6. Given one angle of a triangle, the side opposite to it, and the sum of the remaining two sides. 7. Through a given point A outside a given circle with center O, construct a line to intersect the circle at B and C so that the triangle OBC is greatest possible. 8. Inscribe a square in a given rhombus. 9. Construct a regular octagon whose vertices lie on the sides of a given square. 10. Construct a circle tangent to a given circle and passing through two external points. 220 Euclid’s Elements, Book IV Chapter 7

Modern reorganization of Euclid’s Books III and IV

Theorem 42 (Euclid III.3) The line joining the center of a circle to the midpoint of a chord (which is not a diameter) is perpendicular to the chord.

Theorem 43 (Euclid III.3) The perpendicular from the center of a cir- cle to a chord bisects the chord.

Theorem 44 (Euclid III.14) C Equal chords of a circle are equidistant from the center. N Theorem 45 (Euclid III.14) O

Chords of a circle equidistant from the cen- D ter are equal in length. A B M 222 Modern reorganization of Euclid’s Books III and IV

Theorem 46 (Euclid IV.5) A Given three points not on the same line, there is a unique circle passing through them. O Abbreviation: circumcircle theorem. B C

Theorem 47 (Euclid III.20) The angle which a minor arc of a circle subtends at the center is twice of the angle it subtends at any point on the complementary major arc.

P P

O O P O

A B A B A B

Theorem 48 (Euclid III.21) Angles in the same segment of a circle are equal.

P P Q P

A B B

O Q O O

A B A

Theorem 49 The angle in a semicircle is a right angle. 223

Theorem 50 (Euclid III.22) (a) The opposite angles of a cyclic quadrilateral are supplementary. (b) If one side of a cyclic quadrilateral is extended, the exterior angle so formed is equal to the interior opposite angle.

P P

O O

A B A B

Q Q

Theorem 51 The circle described on the hypotenuse of a right-angled triangle as diameter passes through the vertex of the right angle.

Theorem 52 (Euclid III.20) If a segment subtends equal angles at two points on the same side, the four points are concyclic.

Q P P

O O

A B A B

Q

Theorem 53 If a pair of opposite angles of a quadrilateral are supplementary, the vertices are concyclic. 224 Modern reorganization of Euclid’s Books III and IV

Theorem 54 (Euclid III.26) (a) In equal circles (or the same circle), equal angles at the centers (or centter) stand on equal arcs. (b) In equal circles (or the same circle), equal angles at the circumferences (or cir- cumference) stand on equal arcs.

Theorem 55 (Euclid III.27) (a) In equal circles (or the same circle), equal arcs subtend equal angles at the centers (or centter). (b) In equal circles (or the same circle), equal arcs subtend equal angles at the circumferences (or circumference).

Theorem 56 (Euclid III.28) In equal circles (or the same circle), equal chords cut off equal arcs.

Theorem 57 (Euclid III.29) In equal circles (or the same circle), chords of equal arcs are equal. 225

Theorem 58 (Euclid III.16 Porism) The line perpendicular to a radius of a cir- cle at its extremity is tangent to the circle. O

Theorem 59 (Euclid III.18)

A tangent to a circle is perpendicular to the A radius through the point of tangency.

Theorem 60 P If two tangents are drawn to a circle from an external point, (a) the tangents are equal, (b) the line joining the point to the center bisects the angle between the two tangents. O

Theorem 61 (Euclid III.32) The angle between a tangent to a cir- cle and a chord of the circle through the point of tangency is equal to the O angle subtended by the chord at any point on the opposite arc of the cir- cle.

Theorem 62 A line through an endpoint of a chord of a circle making the same angle with the chord as the angle subtended at any point on the opposite arc of the circle is tangent to the circle.

Theorem 63 (Euclid III.11,12) If two circles are tangent to each other, the line joining their centers passes through the point of tangency.

  O P O O O P 226 Modern reorganization of Euclid’s Books III and IV

Basic Construction 10 (Euclid III.16 Porism) To construct the tangent to a circle at a point on the circumference.

Basic Construction 11 (Euclid III.17) To construct the tangents to a circle from a point outside the circumference.

Basic Construction 12 To construct the exterior common tangents of two circles.

Basic Construction 13 To construct the interior common tangents of two circles.

Basic Construction 14 (Euclid IV.4) To construct the incircle of a triangle.

Basic Construction 15 To construct an excircle of a triangle.

Basic Construction 16 (Euclid III.33) On a given line segment, to construct a segment of a circle containing an angle equal to a given angle. 227

Theorem 64 (Euclid II.12) In an obtuse angled triangle, the square on the side opposite to the obtuse angle is equal to the sum of the squares on the sides containing it, plus twice the rectangle contained by one of these sides and the projection of the other side on it.

A A

c b c b

B a C X B X a C

Theorem 65 (Euclid II.13) In any triangle, the square on the opposite side of an acute angle is equal to the sum of the squares on the sides containing it, minus twice the rectangle contained by one of these sides and the projection of the other side on it.

Theorem 66 (Apollonius Theorem) In any triangle, the sum of the squares on two sides is equal to twice the square on half of the third side, plus twice the square of the median which bisects the third side. 228 Modern reorganization of Euclid’s Books III and IV

Theorem 67 (Euclid III.35) (Intersecting chords theorem) If two chords AB and CD of a circle intersect at a point P inside the circle, then area of the rectangle formed by AP and BP is equal to that of the rectangle formed by CP and DP. C C D

O O P

P B B

A D A T

Theorem 68 (Euclid III.36,37) (Intersecting chords theorem) If two chords AB and CD of a circle intersect at a point P outside the circle, then the area of the rectangle formed by AP and BP is equal to that of the rectangle formed by CP and DP, and also equal to the square of the tangent from P to the circle.

Theorem 69 If two segments AB and CD are divided, both internally or both externally, at a point P such that the rectangle formed by AP and BP has the same area as that formed by CP and DP, then the four points A, B, C, D are concyclic. C C D

O O P

P B B

A D A

Basic Construction 17 (Euclid II.14) To construct a square equal in area to a given rectangle.

Basic Construction 18 To construct a circle passing through two given points and tangent to a given line (not containing the given points).