128. CONSTRUCTION. To divide a given line AB into any

From A draw any indefinite line AB' and lay off upon it any length AC.

Apply AC the required number of times on AB' and sup pose X to be the last point of division. Join XB.

From the various points of division draw parallels to XB. These parallels will cut AB in the required points of division.

Prove this method correct by Proposition XXXIX.

PROBLEMS

129. Exercise.-A straight line parallel to the base of a triangle and bisecting one side bisects the other also.

Hint.-Apply § 127.

А

130. Exercise-A straight line joining the middle points of two sides of a triangle is parallel to the third side.

A

Hint.-Show that this line coincides with a line drawn as in § 129. 131. Exercise.—A straight line joining the middle points of two sides of a triangle equals half the third side.

X

Hint.-Prove DE=BX, and DE=XC.

132. Defs.-A trapezoid is a quadrilateral, two of whose sides are parallel.

The parallel sides are called the bases.

133. Exercise.—A straight line parallel to the bases of a trapezoid and bisecting one of the remaining sides bisects the other also.

134. Exercise.-A straight line joining the middle points of the two non-parallel sides of a trapezoid is parallel to the bases.

135. Exercise.—A straight line joining the middle points of the two non-parallel sides of a trapezoid equals half the sum of the bases.

Hint.-Draw a diagonal and apply § 131.

136. Exercise.-The bisectors of two supplementary-adjacent angles are perpendicular.

137. Exercise.-Any side of a triangle is greater than the difference of the other two.

138. Exercise.-The sum of the three lines from any point within a triangle to the three vertices is less than the sum of the three sides, but greater than half their sum.

Hint.-Apply §§ 7 and 95.

139. Exercise. If from a point in the base of an isosceles triangle parallels to the sides are drawn, a parallelogram is formed, the sum of whose four sides is the same wherever the point is situated (and is equal to the sum of the equal sides).

140. Exercise.—If from a point in the base of an isosceles triangle perpendiculars to the sides are drawn, their sum is the same wherever the point is situated (and is equal to the perpendicular from one extremity of the base to the opposite side).

141. Exercise.-If from a point within an equilateral triangle perpendiculars to the three sides are drawn, the sum of these lines is the same wherever this point is situated (and is equal to the perpendicular from any vertex to the opposite side).

Hint.-Apply § 140.

A

142. Exercise.-The straight lines joining the middle points of the adjacent sides of any quadrilateral form a parallelogram.

Hint.-Apply § 130.

143. Def.—A median of a triangle is a straight line from a vertex to the middle point of the opposite side.

144. Exercise.-The three medians of any triangle intersect in a common point which is two-thirds of the distance from each vertex to the middle of the opposite side.

Hint.-Two of these lines, CE and BD, meet at some point O.
Take M and N, the middle points of BO and CO.

Draw EDNM. Prove it is a parallelogram by proving ED and MN each parallel to and equal to half of BC.

Then prove OE=ON=NC, and DO=OM=MB.

Thus we have proved that one of the medians, as BD, is cut by another, CE, at a point two-thirds of its length from B. We may likewise prove that it is also cut by the third median in the same point.

Hence, etc.