PROPOSITION XXXIV. THEOREM.

118. Of two angles of a triangle, that is the greater which is opposite the greater side.

AL

In the triangle ABC let A B be greater than A C.
We are to prove ZACB> B.

Take A E equal to AC;

Draw EC.
ZAEC = LACE,

§ 112
(being & opposite equal sides).
But LAEC > LB,

§ 105 (an exterior 2 of a A is greater than either opposite interior 2),

ZACB > LACE.
Substitute for Z A C E its equal Z A E C, then

ZACB>ZA EC.
Much more is 2 ACB> B.

and

Q. E. D.

Ex. If the angles A B C and AC B, at the base of an isosceles triangle, be bisected by the straight lines BD, CD, show that DBC will be an isosceles triangle.

PROPOSITION XXXV. THEOREM. 119. The three bisectors of the three angles of a triangle meet in a point.

Cons.

Let the two bisectors of the angles A and C meet

at 0, and 0 B be drawn.
We are to prove BO bisects the 2 B.

Draw the Is OK, O P, and 0 H.
In the rt. A OC K and OCP,
0 C = 0 C,

Iden.
ZOCK = LOCP,
. AOC K= A OCP,

$ 110 (having the hypotenuse and an acute 2 of the one equal respectively to the

hypotenuse and an acute of the other).

..OP=OK,

(homologous sides of equal A).
In the rt. A OA P and 0 A H,
OA = 0 A,

Iden.
ZOAP=20 A H,

Cons. .. AO AP=A O A H,

§ 110 (having the hypotenuse and an acute L of the one equal respectively to the

hypotenuse and an acute Z of the other).

..OP=0H,
(being homologous sides of equal A ).
But we have already shown 0 P = 0 K,
..OH = OK,

Ax. 1
Now in rt. A OH B and 0 KB

OH = 0 K, and 0 B = 0 B,
..A OHB= A 0 KB,

$ 109 (having the hypotenuse and a side of the one equal respectively to the hypote

nuse and a side of the other),
ii ZOBH= LOBK,
(being homologous & of equal A ).

Q. E. D.

PROPOSITION XXXVI. THEOREM. 120. The three perpendiculars erected at the middle points of the three sides of a triangle meet in a point.

Let D D', E E, F F', be three perpendiculars erected

at D, E, F, the middle points of A B, A C, and BC. We are to prove they meet in some point, as 0.

The two is D D' and E E' meet, otherwise they would be parallel, and A B and A C, being is to these lines from the same point A, would be in the same straight line;

but this is impossible, since they are sides of a A.
Let O be the point at which they meet.

Then, since O is in D D', which is I to A B at its middle point, it is equally distant from A and B.

§ 59 Also, since () is in E E', I to A C at its middle point, it is equally distant from A and C.

..O is equally distant from B and C;

..O is in FF I to B C at its middle point, $ 59 (the locus of all points equally distant from the extremities of a straight line is the I erected at the middle of that line).

Q. E. D.

PROPOSITION XXXVII. THEOREM.

121. The three perpendiculars from the vertices of a triangle to the opposite sides meet in a point.

ВІ

In the triangle ABC, let B P, A H, CK, be the per

pendiculars from the vertices to the opposite
sides.
We are to prove they meet in some point, as 0.
Through the vertices A, B, C, draw

A' B' || to BC,
A' C' || to AC,

B'C' || to A B.
In the A A B A and A B C, we have
AB= AB,

Iden.
ZABA' = Z BAC,

§ 68 (being alternate interior ), Z BA A' = Z A B C.

§ 68 .. A A BA' = A A BC,

§ 107 (having a side and two adj. s of the one equal respectively to a side and

two adj. e of the other).

.. A' B = A C,
(being homologous sides of egual A).

In the ACBC and A B C,

BC= BC,

Iden. Z C BC= _ BCA,

§ 68 (being alternate interior &). < BCC' = Z CBA.

§ 68 .. ACBC" = A ABC,

§ 107 (having a side and two adj. of the one equal respectively to a side and two

adj. As of the other).

..BC = AC,
(being homologous sides of equal ).
But we have already shown A' B=AC,

.. AB=BC",

Ax. 1.

.. B is the middle point of A' C'.

Since B P is I to A C,

Hyp.

it is I to A' C',

§ 67 (a straight line which is I to one of two lls is I to the other also).

But B is the middle point of A' C';

.:. B Pis I to A' C' at its middle point. In like manner we may prove that

A H is I to A' B' at its middle point,

and C K I to B'C' at its middle point. .. BP, A H, and C K are Is erected at the middle points of the sides of the A A'B'C'.

.. these Is meet in a point.

$ 120 (the three Is erected at the middle points of the sides of a A meet in a point).

Q. E. D.