## ELEMENTARY SYNTHETIC GEOMETRY OF THE POINT, LINE AND CIRCLE IN THE PLANE |

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Page 14

Thus, if B moves into the position of C, the angle between EA and EC is greater

than the angle between EA and E.B.

one another, the angle at E becomes smaller ; and if the stars become coincident,

...

Thus, if B moves into the position of C, the angle between EA and EC is greater

than the angle between EA and E.B.

**Similarly**, if the stars appear to approachone another, the angle at E becomes smaller ; and if the stars become coincident,

...

Page 16

one arm in common lying between the remaining arms, the angles are adjacents

angles. 36°. Def-A radius vector which starts from any given direction and ...

**Similarly**, AOP=2 AOP' – A POP'. /)es.--When two angles, as AOP and POP', haveone arm in common lying between the remaining arms, the angles are adjacents

angles. 36°. Def-A radius vector which starts from any given direction and ...

Page 27

QP = PB – QB, A C B Ot" PB = QP+QB, which is not true. (25°, Ax.) Therefore the

right bisector of AB does not cut AP ; and

passes through P, or P is on the right bisector. g.e.d. This form of proof should be

...

QP = PB – QB, A C B Ot" PB = QP+QB, which is not true. (25°, Ax.) Therefore the

right bisector of AB does not cut AP ; and

**similarly**it does not cut BP; therefore itpasses through P, or P is on the right bisector. g.e.d. This form of proof should be

...

Page 30

... ABF and CGF have BF = FG, (construction) AF = FC, (55°) and Z_BFA =

A_GFC. (40) .. /\ABF =/\CGF, (52) and 4. FCG = A BAC. (52°, Cor.) But 4 ACE is

greater than 4 FCG. e 4 ACE is > / BAC.

/ ACE.

... ABF and CGF have BF = FG, (construction) AF = FC, (55°) and Z_BFA =

A_GFC. (40) .. /\ABF =/\CGF, (52) and 4. FCG = A BAC. (52°, Cor.) But 4 ACE is

greater than 4 FCG. e 4 ACE is > / BAC.

**Similarly**, 4 BCD is >4 ABC, and A BCD =/ ACE.

Page 37

Proof–The As POA and POB are congruent, since they have two sides and an

angle opposite the longer equal in each (65°, 1); . . 4 POA=4|POB, and PO

bisects the 4 AOB.

THREE OR ...

Proof–The As POA and POB are congruent, since they have two sides and an

angle opposite the longer equal in each (65°, 1); . . 4 POA=4|POB, and PO

bisects the 4 AOB.

**Similarly**, if the perpendiculars from Q upon OA and OBTHREE OR ...

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### Common terms and phrases

ABCD algebraic altitude base becomes called centre centre-line chord circle coincide collinear common concurrent congruent considered constant construct Converse corresponding cuts denote describe determine diagonals diameter difference direction distance divided draw drawn equal expressed external figure fixed point four geometric given given line given point gives greater harmonic Hence internal intersect inverse joins length lies line-segment mean measure median meet middle point opposite opposite sides orthogonally pair parallel passes pencil perpendicular perspective placed plane polar polygon position Proof proved quadrangle radical axis radius range ratio reciprocal rectangle regular relation remaining respect right angle right bisector rotation segment sides similar Similarly similitude square straight symbol taken tangent theorem touch triangle vertex vertices

### Popular passages

Page 176 - The square described on the hypothenuse of a rightangled triangle is equal to the sum of the squares described on the other two sides.

Page 183 - The sides of a triangle are proportional to the sines of the opposite angles.

Page 260 - Three lines are in harmonical proportion, when the first is to the third, as the difference between the first and second, is to the difference between the second and third ; and the second is called a harmonic mean between the first and third. The expression 'harmonical proportion...

Page 19 - Every circumference of a. circle, whether the circle be large or small, is supposed to be divided into 360 equal parts called degrees. Each degree is divided into 60 equal parts called minutes, and each minute into 60 equal parts called seconds.

Page 77 - J_ to the given line. .'. the construction gives the perpendicular to a given line at a given point in the line.

Page 122 - And conversely, if the square on one side of a triangle is equal to the Bum of the squares on the other two sides, the angle contained by these two sides is a right angle.

Page 124 - The difference of the squares of two sides of a triangle is equal to the difference of the squares of the segments made by the altitude upon the third side.