For if the line is not parallel to the third side, suppose a line drawn through D, the middle point of 4C, parallel to AB. By § 158, it will pass through E, the middle point of BC, and we shall have two straight lines drawn between the same two points, which by Ax. 12 is impossible. Consequently, the line joining the middle points of two sides of a triangle is parallel to the third side.

Proposition XLIII

160. Draw a trapezoid and a line connecting the middle points of the non-parallel sides. What is the direction of this line with reference to the bases of the trapezoid? How does it compare in length with the sum of the bases?

Theorem. The line which joins the middle points of the non-parallel sides of a trapezoid is parallel to the bases and is equal to one half their sum.

Data: Any trapezoid, as ABCD, and the line EF joining the middle points of the non-parallel sides AD and BC.

To prove EF parallel to AB and DC and equal to one half AB + DC.

A

E

H

B

Proof. Draw AC intersecting EF at K, and from H, the middle point of AC, draw HE and HF.

:. § 70,

But, data,

then,

and

EHF is a straight line parallel to AB and DC.

EKF is a straight line,

EHF and EKF coincide,

the point H coincides with the point K.

POLYGONS

161. A portion of a plane bounded by any number of straight lines is called a Polygon.

The sum of the straight lines which bound a polygon is called its perimeter.

The term polygon is usually applied to figures of more than four sides.

162. A polygon of three sides is called a trigon or triangle; one of four sides, a tetragon or quadrilateral; one of five sides, a pentagon; one of six sides, a hexagon; one of seven sides, a heptagon; one of eight sides, an octagon; one of ten sides, a decagon; one of twelve sides, a dodecagon; one of fifteen sides, a pentadecagon.

163. A polygon such that none of its sides, if produced, extend within it is called a Convex Polygon.

164. A polygon such that two or more of its sides, if produced, extend within it is called a A Concave Polygon.

The reflex angle ABC is called a re-entrant angle. When the term polygon is used, a convex polygon is meant.

Diagonal

B

165. A straight line joining the vertices of two non-adjacent angles of a polygon is called a Diagonal of the Polygon.

Proposition XLIV

166. 1. Draw convex polygons, each having a different number of sides, and from any vertex of each draw its diagonals. How does the number of triangles into which each polygon is divided compare with the number of sides of the polygon?

To how many right angles is the sum of the angles of a triangle equal? To how many times two right angles is the sum of the interior angles of a polygon equal?

To how many

2. Produce the sides of any polygon in succession. right angles is the sum of all the exterior and interior angles equal? To how many right angles is the sum of the exterior angles of a polygon equal?

Theorem. The sum of the angles of any convex polygon is equal to twice as many right angles as the polygon has sides less two.

Data: A convex polygon of any number (n) of sides, as ABCDE.

To prove the sum of the angles, A, B, C, D, and E equal to twice as many right angles as the polygon has sides less two.

Proof.

E

A

Ꭰ

B

C

From any vertex, as 4, draw the diagonals, AC and AD. The number of triangles thus formed is two less than the number of sides of the polygon, or (n − 2) triangles.

By § 110, the sum of the angles of each triangle is equal to two right angles, therefore, the sum of the angles of all the triangles; that is, the sum of the angles of the polygon is equal to (n-2) 2 rt. 4.

Therefore, etc.

167. Cor. The sum of the exterior angles of any convex polygon formed by producing the sides of the polygon in succession is equal to four right angles.

Ex. 106. If from the extremities of the shorter base of an isosceles trapezoid lines are drawn parallel to the equal sides, two triangles are formed. How do they compare?

Ex. 107. If in a parallelogram any two points in a diagonal equally distant from its extremities are joined to the vertices of the opposite angles, what kind of a figure is thus formed?

Ex. 108. How many degrees are there in each angle of an equiangular polygon of five sides?

Ex. 109. How many sides has a polygon the sum of whose interior angles is double the sum of its exterior angles?

Proposition XLV

168. Draw any triangle and its three medians. Do the medians intersect in a point? Measure the distance from this point to each vertex. How do these distances compare with the medians of which they are a part?

Theorem. The medians of a triangle pass through a point which is two thirds of the distance from each vertex to the middle of the opposite side.

Data: Any triangle, as ABC, and its medians, AD, BE, and CF.

To prove that AD, BE, and CF pass through a point, which is two thirds. of the distance from A, B, and C to the middle of the opposite sides respectively.

A

E

H

K

F

D

Proof. Since two of the medians will intersect, if sufficiently produced, it needs to be shown only that the third median passes through the point of intersection, to prove that the three pass through the same point.

Let any two of the medians, as AD and BE, intersect at H. Draw KG, joining K and G, the middle points of AH and BH respectively; also draw KE, ED, and GD.

Then, since the medians from any two vertices intersect in a point which is two thirds of the distance from each vertex to the middle of the opposite side, the median from Cintersects AD at H. CF passes through H, and CH=CF.

That is,
Therefore, etc.

Q.E.D.

Proposition XLVI

169. Draw any triangle and lines bisecting its angles. Do these lines intersect in a point? How do the distances of the point from the sides of the triangle compare?

Theorem. The bisectors of the three angles of a triangle pass through a point which is equidistant from the sides of the triangle.

Data: Any triangle, as ABC, and the lines AD, BE, and CF, bisecting the angles A, B, and C respectively.

To prove that AD, BE, and CF pass through a point which is equidistant from AB, BC, and AC.

A

E

H

F

B

Proof. Since two of the bisectors will intersect, if sufficiently produced, it needs to be shown only that the third bisector passes through the point of intersection to prove that the three pass through the same point.

Let any two of the bisectors, as AD and BE, intersect in H. Then, § 134, H is equidistant from AB and AC, and also from AB and BC.

Hence, .. § 135,

H is equidistant from AC and BC;

H lies in the bisector of angle C.

That is, CF passes through the point H, which is equidistant from AB, BC, and AC.

Therefore, etc.

Q.E.D.

Ex. 110. How does the angle formed by the diagonals of a square compare with a right angle?

Ex. 111. How does the angle formed by the diagonals of a rhombus compare with a right angle? How do the diagonals divide each other?

Ex. 112. How do the diagonals of a rectangle compare in length?

Ex. 113. If from any point in the bisector of an angle straight lines are drawn parallel to the sides of the angle and are produced to meet the sides, what figure is thus formed?

Ex. 114. The difference between two angles of a parallelogram which have a common side is 60°. What is the value of each angle of the parallelogram? Ex. 115. If the middle points of any two opposite sides of a quadrilateral are joined to each of the middle points of the diagonals, what kind of a figure will the four joining lines forin?