PROPOSITION XXXVI. THEOREM

217 The sum of the exterior angles of a polygon, formed by producing one side at each vertex, is equal to four right angles.

At each vertex of the polygon there are one interior and one exterior angle whose sum is 2 rt. .

§ 103 .. the sum of all the interior and exterior angles of the polygon is 2 n rt. 4. Subtracting the sum of the interior 4, 2 n rt.

- 4rt. 4, § 215

4 rt. s remain

as the sum of the exterior of the polygon.

Q. E. D.

218 COROLLARY. Each exterior angle of an equiangular

4

polygon of n sides is equal to right angles.

n

EXERCISES

145 Prove Prop. XXXVI by drawing from a given point lines parallel to the sides of the polygon.

146 How many sides has a polygon if the sum of the interior angles is eight times the sum of the exterior angles?

147 How many sides has an equiangular polygon if three of its exterior angles are together equal to two right angles?

148 The bisectors of the angles of a triangle meet in a point which is equidistant from the sides of the triangle.

PROOF. Let the angle-bisectors BE and CF intersect in O. Then O is equidistant from AB, AC, and BC (§ 192); and being equidistant from AB and AC, O is in the bisector of the ▲A (§ 193). That is, the three anglebisectors meet in O.

F

DEFINITION. O is called the incenter of the triangle ABC.

D

E

149 The perpendicular bisectors of the sides of a triangle meet in a point which is equidistant from the vertices of the triangle. PROOF. Let the bisectors FF' and EE' intersect in O. Then O is equidistant from A, B, and C (§ 186); and being equidistant from B and C, O is in the bisector of BC (§ 187). That is, the three bisectors meet in O. DEFINITION. O is called the circumcenter of the triangle ABC.

B

D'

E

10

D

K

150 The altitudes of a triangle meet in a point. PROOF. Let AD, BE, and CF be the altitudes of the AABC. Through the vertices of the ▲ draw ||s to the opposite sides, intersecting in G, H, and K. Then BCAG and BCKA are by const. ; whence GA = BC = AK (§ 200). .. AD is the bisector of GK (§ 117). Likewise, BE and CF are the 1 bisectors of GH and HK respectively. That is, the three altitudes meet in O (Ex. 149).

DEFINITION. O is called the orthocenter of the triangle ABC.

151 The medians of a triangle meet in a point of trisection. PROOF. Let the medians BE and CF intersect in O. Draw BH || to FC meeting AOD produced in H, and join HC. Then in the ▲ ABH, AO = OH (§ 209); whence OE is to HC (§ 210). ... BOCH is a □, and BD = DC (§ 206). ... AOD is the third median. That is, the three medians meet in O. Again, OD = OH (§ 206) = AO (Ax. 9) = } AD. Also, OF HB (§ 210) = CO (Ax. 9) Likewise OE = } BE.

B

DEFINITION. O is called the centroid of the triangle ABC.

152 In what kind of a triangle is the circumcenter within the triangle? without the triangle? in one side of the triangle?

153 In what kind of a triangle does the orthocenter fall within the triangle? without the triangle? coincide with one vertex of the triangle?

154 Is the centroid always within the triangle ?

155 In what kind of a triangle do one altitude and one median only coincide?

156 In an equilateral triangle, the incenter, the circumcenter, the orthocenter, and the centroid coincide.

157 In the figure of Ex. 151, prove that the triangles ABC and DEF are mutually equiangular.

158 In the figure of Ex. 151, prove that O is the centroid of the triangle DEF. [Join FE, FD, ED, and let AD intersect FE at G. AEDF is a (§ 210). ... AD bisects FE at G (§ 206). That is, in the A DEF, DG is the median to FE. In like manner show that the medians to DF and DE fall on EB and FC respectively.]

159 In the figure of Ex. 150, prove that the triangle GHK is four times the triangle ABC.

160 Why do the perpendicular bisectors of two sides of a triangle intersect? [Show that they are not parallel.]

161 The six angles formed about the orthocenter of an equilateral triangle are equal.

162 In the triangle ABC, prove that the bisectors of the angle A and the exterior angles B and C meet in a point.

163 If a diagonal of a parallelogram bisects the angles, the parallelogram is equilateral.

B

164 If the diagonals of a parallelogram are perpendicular to each other, the parallelogram is equilateral.

165 If the diagonals of a parallelogram are equal, the figure is a rectangle.

166 If the diagonals of a parallelogram are equal and bisect the angles, the figure is a square.

167 If the diagonals of a parallelogram are equal and perpendicular to each other, the figure is a square.

REVIEW QUESTIONS

1 What is a proposition?

2 How are propositions distinguished? Define each.

3 What is a demonstration ? a solution?

4 Distinguish between a direct demonstration and an indirect demonstration.

5 Distinguish between the two parts of a theorem.

6 What is the converse of a theorem? the opposite ?

7 State the logical relation existing between a theorem, its converse, and its opposite.

8 Define magnitude; extension; space.

9 Distinguish between a material solid and a geometric solid.

10 How many dimensions has a solid? a surface? a line? a point? 11 What are the boundaries of a solid? of a surface? of a line? 12 How may a line be traced? a surface? a solid ?

13 Give the abstract conception of a point; of a line; of a surface; of a solid.

14 What are the fundamental concepts in Geometry? Define each. 15 Give the classification of lines and define each.

16 Define a plane surface; a curved surface.

17 Define Geometry (§ 56).

18 Give the classification of angles, and define each.

19 What is the complement of an angle ? the supplement?

20 What are perpendicular lines? oblique lines?

21 Define geometric figure; plane figure; rectilinear figure; curvilinear figure.

22 What is the unit of angular measurement? How obtained ?

23 What are the geometric magnitudes ?

24 Distinguish between similar, equivalent, and equal magnitudes.

25 What is the test of equality of geometric magnitudes ?

26 What is the object of Geometry?

27 How is Geometry divided ?

28 Of what does Plane Geometry treat?

29 Can you prove an axiom?

30 Distinguish between an axiom and a postulate.

31 Can you give an axiom or a postulate not found in the text? 32 How many points determine a straight line?

33 What are supplementary adjacent angles ? vertical angles ? 34 What are parallel lines?

35 Define a polygon; the sides, angles, and vertices of a polygon.

36 What are adjacent angles of a polygon?

37 What is an exterior angle of a polygon?

38 Give the classification of triangles, and define each.

39 Define an equiangular triangle; an equilateral triangle. 40 Is an equilateral triangle equiangular? why ?

41 Is an equiangular triangle equilateral? why ?

42 Can a leg of a right triangle be equal to the hypotenuse?

43 What is the base of a triangle ?

44 What is the base of an isosceles triangle ?

45 What is the vertex of a triangle ?

46 Define a median, an altitude, an angle-bisector of a triangle.

47 Define mutually equilateral polygons; mutually equiangular polygons.

48 What are homologous lines in mutually equiangular polygons ? 49 Define quadrilateral; trapezium; trapezoid.

50 What does the word "parallelogram

99 mean?

51 Define rectangle; square; rhomboid; rhombus.

52 Define the bases of a trapezoid; the median; the legs.

53 What is the altitude of a parallelogram or trapezoid ?

54 What does the word "polygon" mean?

55 Define pentagon; hexagon; heptagon; octagon; decagon; dodecagon; pentadecagon.

56 Define equilateral polygon; equiangular polygon; convex polygon; concave polygon.

57 How may any polygon be divided into triangles ?

58 Write a short essay on the great geometers Thales and Pythagoras.