To prove that ABC is equivalent to the rectangle of AB, CD. Complete the parallelogram ABEC.

The triangle ABC is equivalent to half the parallelogram ABEC (I. 153); and therefore equivalent to half the rectangle of AB and CD (25).

Ex. Prove this theorem directly by applying 14, 17 to the adjoining figures.

30. Cor. I. A trapezoid is equivalent to the rectangle contained by its altitude and half the sum of its parallel sides.

31. Cor. 2. If two triangles have equal altitudes, then according as the base of the first is greater than, equal to, or less than the base of the second, so is the first triangle greater than, equivalent to, or less than the second. (Use 23, 29.)

32. Cor. 3. If two triangles have equal altitudes, then according as the first triangle is greater than, equivalent to, or less than the second, so is the base of the first greater than, equivalent to, or less than the base of the second.

Ex. If there are two equilateral triangles, then according as a side of the first is greater than, equal to, or less than a side of the second, so is the first triangle greater than, equal to, or less than the second.

33. Cor. 4. Two triangles having the same base, and having their opposite vertices on the same line parallel to the base, are equivalent. Conversely, two equivalent triangles on the same base and at the same side of it are between the same parallels.

34. Cor. 5. If a parallelogram and a triangle are upon the same base and between the same parallels, the parallelogram is double the triangle.

Parallelograms about a diagonal.

35. Definition. If through any point on the diagonal of a parallelogram two lines be drawn parallel to the sides, so as to divide the parallelogram into four smaller parallelograms, the two whose diagonals are portions of the diagonal first mentioned are called the parallelograms about the diagonal; and the two which lie one on each side of the diagonal are called the complements of the parallelograms about the diagonal.

36. THEOREM 9. The complements of the parallelograms about the diagonal of a parallelogram are equivalent.

Let ABCD be the parallelogram, BD its diagonal, K any point on it, FH and EG lines through K parallel to the sides, forming KGBF and DHKE parallelograms about the diagonal, and KFAE and CGKH the complements of these parallelograms.

To prove that these complements are equivalent.

Since the diagonal of a parallelogram bisects it (I. 153), the triangle DBA is equivalent to CBD; similarly KBF is equivalent to KGB; and KED to DHK.

Take KBF and KED away from DBA, and take KGB and DHK away from CBD; then the remainders KFAE and KHCG are equivalent (15).

NOTE. This theorem is useful in the construction of equivalent parallelograms (72).

37. THEOREM 10. Parallelograms about the diagonal of a rhombus are rhombuses, and their complements are equal parallelograms.

Use 161, 117, 166 of Book I.

38. Cor. Parallelograms about the diagonal of a square are squares, and their complements are equal rectangles.

EXERCISES

1. If one diagonal of a quadrangle bisects the other, it also bisects the quadrangle.

2. If a parallelogram and a triangle are such that the base and altitude of the parallelogram are respectively equal to half the base and altitude of the triangle, then the parallelogram is equivalent to half the triangle.

3. Lines joining the mid-points of adjacent sides of a quadrangle form a parallelogram equivalent to half the quadrangle.

4. If two triangles stand on the same base and at the same side of it, and if the middle points of the sides are joined, then the joining lines form a parallelogram equivalent to half the difference of the triangles. 5. To construct an isosceles triangle equivalent to a given triangle and standing on the same base.

6. To construct a rhombus equivalent to a given parallelogram and having the same diagonal.

7. A triangle whose base is one of the non-parallel sides of a trapezoid and whose vertex is at the mid-point of the opposite side is equivalent to half the trapezoid.

[Through the mid-point in question draw a parallel to the opposite side and complete the parallelogram.]

EQUIVALENCES INVOLVING RECTANGLES

Rectangles of wholes and parts.

39. THEOREM 11. If there are two lines, one of which is divided into any number of parts at given points, the rectangle of the two given lines is equivalent to the sum of the rectangles of the undivided line and the several parts of the divided line.

Let AB, CF be the two lines, and let CF be divided at the points D and E into the parts CD, DE, EF.

To prove that the rectangle of AB and CF is equal to the sum of the rectangles of AB, CD; AB, DE; AB, EF.

Draw the line CG perpendicular to CF and equal to AB. Complete the rectangle CFLG, and draw DH, EK perpendicular to CF.

The lines DH, EK are equal to CG (I. 153) and therefore equal to AB.

The rectangle CL is equivalent to the sum of the rectangles CH, DK, EL.

Now CH is the rectangle of CG and CD, that is, of AB and CD; also DK is the rectangle of AB and DE; and EL is the rectangle of AB and EF.

Therefore the theorem is established.

NOTE. Two of the following corollaries are special cases of this theorem, and the third is an extension of it.

Rectangle of whole line and one part.

40 (a). Cor. I. If a line is divided into any two parts, the rectangle of the whole line and one part is equivalent to the square on that part together with the rectangle of the two parts.

Square on whole line.

40 (b). Cor. 2. If a line is divided into any two parts, the square on the whole line is equivalent to the sum of the rectangles of the whole line and each of the parts.

Distributive property of rectangles.

41. Cor. 3. If each of two lines is divided into any number of parts, then the rectangle contained by the whole lines is equivalent to the sum of all the rectangles contained by each part of one and each part of the other.

[Prove by repeated applications of 39; or else by an independent figure.]

NOTE. This important principle will be referred to as "the distributive property of rectangles"; it lies at the foundation of many of the subsequent theorems.

Ex. 1. Show that 39, 40 are special cases of the "distributive property."

Ex. 2. If a line is divided into three parts, then the square on the whole line is equivalent to the sum of the rectangles of the whole line and each of its parts.