THEOREM XXXII.

98. CONVERSELY.-If two angles of a triangle are equal, the sides opposite them are equal, and the triangle is isosceles.

99. Of two angles of a triangle, the greater is opposite the greater side.

In the ▲ ABC, let

CB > AB.

C

THEOREM XXXIV.

100. CONVERSELY.-Of two sides of a triangle, the greater is opposite the greater angle.

In the ABC, let BAC > < c.

D

B

A

To prove that BC > AB.

Let AD be drawn so as to make a = L c.

BISECTORS OF ANGLES.

THEOREM XXXV.

101. Any point in the bisector of an angle is equally distant from the sides of the angle.

Let BF be the bisector of the ABC, Pany point in it, and PD and PE Ls to AB and BC.

102. COR.-Any point in an angle equally distant from the

sides lies in the bisector of the angle.

THEOREM XXXVI.

103. The bisectors of the angles of a triangle meet in a common point.

Let AE, BF, and CD be the bisectors of the As of the Δ ΑΒΓ.

To prove that AE, BF, and CD meet in a common point.

Let AE and BF meet in a point, as O.

Then is equally distant from AB and AC; also from

AB and BC;

(101)

... O is equally distant from AC and BC, and lies in CD;

(102)

.. the bisectors AE, BF, and CD meet in a common point. Q. E. D.

104. COR.-The point in which the bisectors of the angles of a triangle meet is equally distant from the three sides of the triangle.

THEOREM XXXVII.

105. The perpendiculars erected at the middle points of the sides of a triangle meet in a common point.

In the ▲ ABC, let DH, FG, and EM be respectively to AC, AB, and BC, at their middle points.

To prove that DH, FG, and EM meet at a common point.

The Ls DH and FG meet in some point, as O, otherwise they would be , and AC and AB, the Ls to these Ils, would lie in the same straight line, which is impossible.

Now is equally distant from A and C; also from A and B;

(53)

... O is equally distant from C and B, and must lie in EM (54). That is, the EM passes through 0;

... DH, FG, and EM meet in a common point. Q. E. D.

106. COR.-The common point of the perpendiculars erected at the middle points of the sides of a triangle is equally distant from the vertices of the triangle.