through the vertex B [6], and the points A and Care symmetric to DE [I. 29], the lines A B and CB are symmetric to DE; threefore A B is equal to B C, which W. T. B. D. Inversely The angles opposite to two unequal sides of a triangle, are unequal to each other.

COROLLARY I. An equiangular triangle is equilateral.

COROLLARY II. The angles of a scalene triangle,

are unequal to each other.

THEOREM 8.

(Eucl. I. 5.) The angles at the base of an isosceles triangle, are equal to one another.

Let the sides A B and BC of the triangle A B C be equal to each other.

Let B D be the bisectrix of the angle ABC; then the vertices A and C, are symmetric points of the angle ABC. Because AB and C B are

A

B

P

symmetric to B D [II. 6 ii], and the two portions A D and D C of the base, are symmetric to the same line [I. 27 ii], the angles B A D and BCD have their sides symmetric to BD: therefore they are equal to each other [II. 19], W. W. T. B. D. Inversely

The sides opposite to unequal angles of a triangle, are unequal. COROLLARY I. All the interior, or exterior, angles of an equilateral triangle, are equal to one another.

COROLLARY II. If the equal sides of an isosceles triangle be produced beyond the base, they form with the base two equal exterior angles.

COROLLARY III. If the angles of a triangle be unequal, the triangle is scalene.

COROLLARY IV. The altitude of an isosceles triangle is the bisectrix of the angle at the apex, and conversely.

COROLLARY V. The altitude of an isosceles triangle divides its perimeter into two symmetric parts.

Scholium. The equal sides of an isosceles right-angled triangle are those which form the right angle (I. 11).

THEOREM 9.

(Eucl. I. 19.)

A greater angle of a triangle is opposite a greater side.

In the triangle A B C, let the angle B A C be greater than the angle B CA.

Let an angle DAC be equal to the angle BCA: then the side

DA, in the triangle A D C, is equal to the side D C [7]. Because. BA is less than B D and D A together [5], B A is less than B D and DC together [Post. II]; that is, B A is less than B C, which W.T. B.D. Inversely

A greater side of a triangle is opposite a greater angle.

Scholium. The greatest angle of a triangle is opposite the greatest side, and the smallest angle opposite the smallest side, and conversely.

THEOREM 10.

In a right-angled triangle, the median from the vertex of the right angle is equal to half the hypothenuse.

Let A B C be a triangle right-angled in B.

Let the angle ABD be equal to the angle A: then the triangle A B D is isosceles [7]; that is, BD is equal to AD. Because the angles DBC and C are complementary to equal angles [3], they are equal to each other [II. 15 i]; consequently the triangle DBC is isosceles [7], and B D is equal to DC: therefore A D is equal to DC [i]; that is, BD is a median of the triangle A B C, and is equal to half of A C, which W. T. B. D. Inversely

An angle of a triangle is not a right angle, if the median from its vertex be not equal to half the opposite side.

COROLLARY I. Conversely, an angle of a triangle is a right angle, if the median through its vertex be equal to half the opposite side.

COROLLARY II. In any triangle, the median through an obtuse angle is less than the opposite side, and the median through an acute angle is greater than the opposite side, and conversely.

THEOREM 11.

The middle line of two sides of a triangle, is equal to half the third side.

Let D E be the middle line of the sides A B and B C of the triangle ABC.

Because the sides of the angle ABC are cut proportionally by DE and A C, the lines D E and AC are parallel to each other [II. 39]: therefore D E is equal to half of A C [II. 41 i], which W. T. B. D. COROLLARY I. The middle line of any two sides of a triangle is parallel to the third side.

COROLLARY II. The median to each side of a triangle bisects the middle line of the other sides.

COROLLÁRY III. The middle lines of the sides of an isosceles or equilateral triangle, divide it into four isosceles or equilateral triangles.

COROLLARY IV. The parallels through the vertices of an isosceles or equilateral triangle, form an isosceles or equilateral triangle.

THEOREM 12.

In every triangle, the middle perpendiculars on the three sides meet at the same point.

Let D E, F H and IK be the middle perpendiculars on the sides of the triangle A B C.

Because A B and B C are not parallel, the lines DE and F H perpendicular thereon will meet, if indefinitely produced [I. 18 ii]. Let O be the intersection of DE and FH. Because the point O belongs to the middle perpendicular on AB, it is equally distant from the vertices A and B [I. 14]; because the same point belongs to the middle perpendicular on B C, it is also equally distant from the vertices B and C ; hence the point O is equally distant from the vertices A and C [i], and belongs to the middle perpendicular on A C [I. 15 i]: therefore I K will meet D E and F H at O, which W. T. B. D.

COROLLARY. The three altitudes of a triangle meet at the same point. Scholium. The middle perpendiculars on the sides of a triangle are the altitudes of the triangle formed by the middle lines of the sides.

THEOREM 13.

In every triangle, the bisectrices of the three angles meet at the same point.