17. If two circumferences cut each other, and from either point of intersection a diameter be made in each, the extremities of these diameters and the other point of intersection are in the same straight line.

18. If any straight line joining two parallel lines be bisected, any other line through the point of bisection and joining the two parallels, is also bisected at that point.

19. If two circumferences are concentric, a line which is a chord of the one and a tangent of the other, is bisected at the point of contact.

20. If a circle have any number of equal chords, what is the locus of their points of bisection?

21. If any point, not the center, be taken in a diameter of a circle, of all the chords which can pass through that point, that one is the least which is at right angles to the diameter.

22. If from any point there extend two lines tangent to a circumference, the angle contained by the tangents is double the angle contained by the line joining the points of contact and the radius extending to one of them.

23. If from the ends of a diameter perpendiculars be let fall on any line cutting the circumference, the parts intercepted between those perpendiculars and the curve are equal.

24. To draw a circumference with a given radius, so that the sides of a given angle shall be tangents to it.

25. To draw a circumference through two given pois, with the center in a given line.

26. Through a given point, to draw a straight line, making equal angles with the two sides of a given angle.

CHAPTER V.

TRIANGLES.

246. Next in regular order is the consideration of those plane figures which inclose an area; and, first, of those whose boundaries are straight lines.

A POLYGON is a portion of a plane bounded by straight lines. The straight lines are the sides of the polygon.

The PERIMETER of a polygon is its boundary, or the sum of all the sides. Sometimes this word is used to designate the boundary of any plane figure.

247. A TRIANGLE is a polygon of three sides.

Less than three straight lines can not inclose a surface, for two straight lines can have only one common point (51). Therefore, the triangle is the simplest polygon. From a consideration of its properties, those of all other polygons may be derived.

248. Problem.—Any three points not in the same straight line may be made the vertices of the three angles of a triangle.

For these points determine the plane (60), and straight lines may join them two and two (47), thus forming the required figure.

INSCRIBED AND CIRCUMSCRIBED.

249. Corollary.-Any three points of a circumference may be made the vertices of a triangle. A circumfer

ence may pass through the vertices of any triangle, for it may pass through any three points not in the same straight line (149).

250. Theorem. Within every triangle there is a point equally distant from the three sides.

In the triangle ABC, let lines bisecting the angles A and B be produced until they

A

the line which bisects the angle B (113). For a similar reason, the point D is equally distant from the two sides AB and AC. Therefore, it is equally distant from the three sides of the triangle.

251. Corollary.—The three lines which bisect the several angles of a triangle meet at one point. For the point D must be in the line which bisects the angle C (113).

252. Corollary.-With D as a center, and a radius equal to the distance of D from either side, a circumference may be described, to which every side of the triangle will be a tangent.

253. When a circumference passes through the vertices of all the angles of a polygon, the circle is said to be circumscribed about the polygon, and the polygon to be inscribed in the circle. When every side of a polygon is tangent to a circumference, the circle is inscribed and the polygon circumscribed.

254. The angles at the ends of one side of a triangle are said to be adjacent to that side. Thus, the

A

angles A and B are adjacent to the side AB. The angle formed by the other two sides is opposite. Thus, the angle A and the side BC are opposite to each other.

SUM OF THE ANGLES.

255. Theorem.-The sum of the angles of a triangle is equal to two right angles.

Let the line DE pass through the vertex of one angle, B, parallel to the opposite side, AC.

Then the angle A is equal

to its alternate angle DBA (125). For the same rea

D

A

B

E

son, the angle C is equal to the angle EBC. Hence, the three angles of the triangle are equal to the three consecutive angles at the point B, which are equal to two right angles (91). Therefore, the sum of the three angles of the triangle is equal to two right angles.

256. Corollary.-Each angle of a triangle is the supplement of the sum of the other two.

257. Corollary.-At least two of the angles of a triangle are acute.

258. Corollary.—If two angles of a triangle are equal, they are both acute. If the three are equal, they are all acute, and each is two-thirds of a right angle.

259. An ACUTE ANGLED triangle

is one which has all its angles acute,

as a.

A RIGHT ANGLED triangle has one of the angles right, as b.

a

t

An OBTUSE ANGLED triangle has

one of the angles obtuse, as c.

с

260. Corollary. In a right angled triangle, the two acute angles are complementary (94).

261. Corollary-If one side of a triangle be pro

duced, the exterior angle thus

formed, as BCD, is equal to

B

the sum of the two interior

angles not adjacent to it, as A A

D

and B (256). So much the more, the exterior angle is greater than either one of the interior angles not adjacent to it.

262. Corollary.-If two angles of a triangle are respectively equal to two angles of another, then the third angles are also equal.

263. Either side of a triangle may be taken as the base. Then the vertex of the angle opposite the base is the vertex of the triangle.

The ALTITUDE of the triangle is the distance from the vertex to the base, which is measured by a perpendicular let fall on the base produced, if necessary.

264. Corollary. The altitude of a triangle is equal to the distance between the base and a line through the vertex parallel to the base.

265..When one of the angles at the base is obtuse, the perpendicular falls outside of the triangle.

When one of the angles at the base is right, the altitude coincides with the perpendicular side.

When both the angles at the base are acute, the altitude falls within the triangle.

Let the student give the reason for each case, and illustrate it with a diagram.