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PROPOSITION IV. THEOREM.

53. Two oblique lines drawn from a point in a perpendicular, cutting off equal distances from the foot of the perpendicular, are equal.

Let FC be the perpendicular, and C A and C O two

oblique lines cutting off equal distances from F.

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Fold over C FA, on C F as an axis, until it comes into the plane of C FO.

FA will take the direction of FO,
(since 2C FA = 2CFO, each being a rt. 2).

Point A will fall upon point 0,

(F A = F 0, by hyp.).

.. line CA = line CO,
(their extremities being the same points).

§ 18

Q. E. D.

PROPOSITION V. THEOREM.

54. The sum of two lines drawn from a point to the extremities of a straight line is greater than the sum of two other lines similarly drawn, but included by them.

Let C A and C B be two lines drawn from the point C

to the extremities of the straight line A B. Let O A and O B be two lines similarly drawn, but included by C A and C B.

We are to prove

CA + C B >0 A + 0 B.

§ 18

Produce A O to meet the line C B at E.
Then AC + CE > A 0 + 0 E,

(a straight line is the shortest distance between two points),
and
BE+O E > BO.

§ 18
Add these inequalities, and we have
CA + CE+BE+O E >0 A + 0 E + O B.
Substitute for C E + B E its equal C B,

and take away 0 E from each side of the inequality.
We have CA + CB >0 A + 0 B.

Q. E. D.

PROPOSITION VI. THEOREM. 55. Of two oblique lines drawn from the same point in a perpendicular, cutting off unequal distances from the foot of the perpendicular, the more remote is the greater.

FK \A - B

Let C F be perpendicular to A B, and C K and C H two

oblique lines cutting off unequal distances from F.
We are to prove CH > C K.
Produce C F to E, making FE = C F.

Draw E K and E H.
CH = H E, and CK= KE,

§ 53 (two oblique lines drawn from the same point in a I, cutting off equal dis

tances from the foot of the I, are equal).
But
C II + HE>CK + KE,

§ 54 (The sum of two oblique lines draun from a point to the extremities of a

straight line is greater than the sum of two other lines similarly drawn,
but included by them);

.:. 2 C H > 2 CK;
.. C II > C K.

Q. E. D. 56. COROLLARY. Only two equal straight lines can be drawn from a point to a straight line; and of two unequal lines, the greater cuts off the greater distance from the foot of the perpendicular.

PROPOSITION VII. THEOREM.

57. Two equal oblique lines, drawn from the same point in a perpendicular, cut off equal distances from the foot of the perpendicular.

F KB

Let C F be the perpendicular, and C E and C K be two

equal oblique lines drawn from the point C.

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Fold over CFA on C F as an axis, until it comes into the plane of C FB.

The line F E will take the direction F K,

(2 CFE= ZC F K, each being a rt. 2).
Then the point E must fall upon the point K;

otherwise one of these oblique lines must be more remote from the I,

and .. greater than the other; which is contrary to the hypothesis.

§ 55 ..FE= FK.

Q. E. D.

PROPOSITION VIII. THEOREM.

58. If at the middle point of a straight line a perpendicular be erected,

I. Any point in the perpendicular is at equal distances from the extremities of the straight line.

II. Any point without the perpendicular is at unequal distances from the extremities of the straight line.

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Let P R be a perpendicular erected at the middle oi

the straight line A B, O any point in P R, and Cany point without P R.

Draw 0 A and O B.

We are to prove OA = 0 B.
Since

PA = PB,

OA= 0 B, (two oblique lines drawn from the same point in a I, cutting off equal dis

tances from the foot of the I, are equal).

§ 53

II.

Draw C A and C B.
We are to prove C A and C B unequal.
One of these lines, as C A, will intersect the I.

From D, the point of intersection, draw D B.

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