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

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Page 67

As a line can cut a G) only twice it can

touches a G) cannot cut it. 4. A G) is determined by two points if one of them is a

given point of contact on a given line ; or, only one circle can pass through a

given ...

As a line can cut a G) only twice it can

**touch**a G) only once. 3. A line whichtouches a G) cannot cut it. 4. A G) is determined by two points if one of them is a

given point of contact on a given line ; or, only one circle can pass through a

given ...

Page 69

Two circles can

points of intersection. I 13”. 7%eorem.—The common centre-line of two

intersecting circles is the right bisector of their common chord, O and O' are the

centres of ...

Two circles can

**touch**in only one point. For a point of contact is equivalent to twopoints of intersection. I 13”. 7%eorem.—The common centre-line of two

intersecting circles is the right bisector of their common chord, O and O' are the

centres of ...

Page 71

Three tangents

the Z\A'B'C', O being the centre of the circle, A AOC = 24 A'OC'. Proof.- AC'— BC',

and BAs a CA', (I 14°, Cor. 1) g /\AOC = /\BOC', and /\BOA' = ACOA' A BOC' - 4 ...

Three tangents

**touch**the circle S at the points A, B, and C, and intersect to formthe Z\A'B'C', O being the centre of the circle, A AOC = 24 A'OC'. Proof.- AC'— BC',

and BAs a CA', (I 14°, Cor. 1) g /\AOC = /\BOC', and /\BOA' = ACOA' A BOC' - 4 ...

Page 74

If r, r' be the radii of two circles, and d the distance between them, the circles

common tangent. Prove Ex. 2, 1 16°, by drawing common tangents to the circles

at P ...

If r, r' be the radii of two circles, and d the distance between them, the circles

**touch**when d- rit?'. . Give the conditions under which two circles have 4, 3, 2, or Icommon tangent. Prove Ex. 2, 1 16°, by drawing common tangents to the circles

at P ...

Page 75

If two circles

arcs which subtend equal angles in the two circles. If any two lines be drawn

through the point of contact of two

whose ...

If two circles

**touch**one another, any line through the point of contact determinesarcs which subtend equal angles in the two circles. If any two lines be drawn

through the point of contact of two

**touching**circles, the lines determine arcswhose ...

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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.