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

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Page 3

Two theorems are

of the one are respectively the conclusion and hypothesis of the other. Ex. If an

animal is a horse it has four legs.

horse.

Two theorems are

**converse**to one another when the hypothesis and conclusionof the one are respectively the conclusion and hypothesis of the other. Ex. If an

animal is a horse it has four legs.

**Converse**. If an animal has four legs it is ahorse.

Page 4

EXERCISES. State the contrapositives and the

theorems – I. The sum of two odd numbers is an even number. 2. A diameter is

the longest chord in a circle. 3. Parallel lines never meet. - 4. Every point

equidistant from ...

EXERCISES. State the contrapositives and the

**converses**of the followingtheorems – I. The sum of two odd numbers is an even number. 2. A diameter is

the longest chord in a circle. 3. Parallel lines never meet. - 4. Every point

equidistant from ...

Page 26

4 APC = 4 BPC ; - Therefore the right bisector of the base of an isosceles triangle

is the internal bisector of the vertical angle. And since these two bisectors are one

and the same line the

4 APC = 4 BPC ; - Therefore the right bisector of the base of an isosceles triangle

is the internal bisector of the vertical angle. And since these two bisectors are one

and the same line the

**converse**is true, Des. 2.-A triangle in which all the sides ... Page 27

... all the angles of an equilateral triangle are equal to one another ; or, an

equilateral triangle is equiangular. 54". Theorem,--Every point equidistant from

the endpoints of a line-segment is on the right bisector of that segment. (

... all the angles of an equilateral triangle are equal to one another ; or, an

equilateral triangle is equiangular. 54". Theorem,--Every point equidistant from

the endpoints of a line-segment is on the right bisector of that segment. (

**Converse**of 53°.) ... Page 28

56°. Theorem.–If two angles of a triangle are equal to one another, the triangle is

isosceles, and the equal sides are P opposite the equal angles. (

, Q Cor. I.) A PAB = 4 PBA, then PA = PB. Proof–If P is on the right bisector of AB ...

56°. Theorem.–If two angles of a triangle are equal to one another, the triangle is

isosceles, and the equal sides are P opposite the equal angles. (

**Converse**of 53°, Q Cor. I.) A PAB = 4 PBA, then PA = PB. Proof–If P is on the right bisector of AB ...

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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.