THEOREM XLIX.

143. The perpendiculars drawn from the vertices of a triangle to the opposite sides meet in a common point.

In the ▲ ABC, let AD, BE, and CF be the Ls from the vertices to the opposite sides.

To

prove that AD, BE, and CF meet in a common point.

Through the vertices, let MH, GM, and GH be drawn respectively to AB, BC, and AC.

Likewise we can prove that AD and BE are Ls to GM and GH at their middle points;

.'.

the three s meet in a common point. (105) Q.E.D.

THEOREM L.

144. The medial lines of a triangle meet in a common point.

In the ▲ ABC, let AD, BE, and CF be the medial lines.

To prove that AD, BE, and CF meet in a common point.

Let AD and CF meet at P, and let M and N be the middle points of CP and AP.

Or AD intersects CF at P, a point whose distance from F

equals CF.

Likewise we can prove that BE intersects CF at a point whose distance from F equals CF.

AD, BE, and CF meet in a common point.

Q. E.D.

EXERCISES IN INVENTION.

THEOREMS.

1. The two straight lines which bisect the two pairs of vertical angles formed by two lines are perpendicular to each other.

2. Two equal straight lines drawn from a point to a straight line make equal angles with that line.

3. If the three sides of an equilateral triangle are produced, all the external acute angles are equal, and all the obtuse angles are equal.

4. If the equal angles of an isosceles triangle are bisected, the triangle formed by the bisectors and the base is an isosceles triangle.

5. The three straight lines joining the middle points of the sides of a triangle divide the triangle into four equal triangles.

6. If one of the acute angles of a right-angled triangle is double the other, the hypothenuse is double the shortest side.

7. If through any point in the base of an isosceles triangle parallels to the equal sides are drawn, a parallelogram is formed whose perimeter equals the sum of the equal sides of the triangle.

8. If the diagonals of a parallelogram bisect each other at right angles, the figure is either a square or a rhombus.

9. The sum of the four lines drawn to the vertices of any quadrilateral from any point except the intersection of the diagonals, is greater than the sum of the diagonals.

10. The straight lines which join the middle points of the adjacent sides of any quadrilateral, form a parallelogram whose perimeter is equal to the sum of the diagonals of the given quadrilateral.

11. Lines joining the middle points of the opposite sides of any quadrilateral, bisect each other.

12. If the four angles of a quadrilateral are bisected, the bisectors form a second quadrilateral whose opposite angles are supplements of each other.

NOTE. If the figure is a rhombus or a square, there is no second one formed.

PROBLEMS.

1. Show by a diagram that between five points, no three of

which lie in the same straight line,

be drawn connecting the points.

5 X 4

2

straight lines can

2. Between n points, no three of which lie in the same

straight line,

nxin 1)

necting the points.

2

straight lines can be drawn con

3. What is the greatest number of angles that can be formed with four straight lines?

Ans. 24.

4. If the sum of the interior angles of a polygon equals the sum of its exterior angles, how many sides has the polygon?

5. If the sum of the interior angles of a polygon is double the sum of its exterior angles, how many sides has the figure?

6. If the sum of the exterior angles of a polygon is double the sum of its interior angles, how many sides has the figure?