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Ex. 649. If two sides of a triangle equal 15 and 25, respectively, and the projection of 15 upon 25 equals 9, what is the value of the third side.

Ex. 650. Two sides of a triangle are 16 and 12 inches, respectively, and include an angle of 60°. Find the third side.

Ex. 651. Two sides of a triangle are 20 and 30, respectively, and include an angle of 45°. Find the third side.

323. SCHOLIUM. If we consider a projection of one side of a triangle upon another as positive when the projection lies on that line, but as negative when it lies on the prolongation, Props. XXXVIII and XXX become special cases of Prop. XXXVII, and we have always :

ao = b + c - 2 cp. To compute the projections of sides of a triangle whose angles are not known, always apply this equation. If the result is negative, the triangle is obtuse.

324. Cor. 1. In Aabe, if pe denotes the projection of b upon c,

62 + 0.2 a Pc

2 c

325. Cor. 2. If he denotes the altitude upon c,

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[This expression can be simplified by algebraical operations: (52 + ? - a' 2

02 + c a 12 + c - a h?=

6
2 c
(2 bc + b3 + c2 a) (2 bc - b02 + a?)

4 c?
[(b + c)2 — a?][a’ – (c)?]

4 ca
(a + b + c)(b + c a)(a b + c)(a + b c).

Let a + b +c= 2 s, i.e. let s denote half the perimeter.

b+c-a= 2 (8 – a).
a-b+c= 2 (8 - b).

a + b c = 2 (8 -c).
h? =48 (– a)(8 b)(8 c)

c2
2
h.
8

c).

or

Ex. 652. In Aabc, a = 20, b 15, and c = 7. Find the projection of b upon c. Is the triangle obtuse or acute ?

Ex. 653. The sides of a triangle are 4, 13, and 15, respectively. Find the three altitudes.

Ex. 654. The sides of a triangle are 13, 14, and 15, respectively. Find the three altitudes.

PROPOSITION XXXIX. THEOREM

326. In any triangle, the square of one side plus four times the square of the corresponding median is equal to twice the sum of the squares of the other sides.

m

D

B

To prove

Нур. In A ABC, m. is the median to c.

c? + 4m? = 2a + 20. Proof. Draw CEI AB, and suppose E to fall between A and D. Let DE=q.

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327. COR. In Aabc, if m, denotes the median drawn to c, m= v2 a' +26.

Ex. 655. The sides of a triangle are 9, 10, and 17, respectively. Find the three altitudes.

Ex. 656. The sides of a triangle are 11, 25, and 30, respectively. Find the three altitudes.

Ex. 657. The sides of a triangle are 7, 8, and 9, respectively. Find the length of the median to 8.

Ex. 658. The sides of a triangle are 7, 4, and 9, respectively. Find the length of the median to 9.

Ex. 659. In A abc, a = 8, b = 11, and mc = 81. Find c.
Ex. 660. In A abc, a = 28, c = 32, and me = 38. Find b.

Ex. 661. The sides of a triangle are 10, 5, and 9, respectively. Find the length of the median to 9.

Ex. 662. The sides of a triangle are 22, 20, and 18, respectively. Find the length of the median to 18.

Ex. 663. The sum of the squares of the four sides of a parallelogram is equal to the sum of the squares of the diagonals.

PROPOSITION XL. THEOREM

328. In any triangle, the product of two sides is equal to the square of the bisector of the included angle plus the product of the segments of the third side.

a

B

Hyp. In A abc, the bisector t divides c into the segments, p and a.

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Proof. Circumscribe a circle about A abc.

Produce the bisector CD to meet the circumference in E. Draw EB, and let DE 2 ACD = 2 ECB.

(Hyp.)

=X.

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t = ab pa.

329. Cor.
By applying (287), it may be proven that

ac

or

P

b + c

bc
9
b + c

abc2
t2 = ab

(b+c)? By using the method and notations of (325), we may obtain

2
t Vab(s – c).

atb

Ex. 664. The sides of a triangle are 18, 9, and 21, respectively. Find the length of the bisector corresponding with 21.

Ex. 665. The sides of a triangle are 21, 14, and 25, respectively. Find the length of the bisector corresponding with 25.

Ex. 666. The sides of a triangle are 22, 11, and 21, respectively. Find the length of the bisector corresponding with 21.

Ex. 667. The sides of a triangle are 6, 3, and 7, respectively. Find the length of the bisector corresponding with 7.

PROPOSITION XLI. THEOREM 330. In any triangle the product of two sides is equal to the altitude upon the third side, multiplied by the diameter of the circumscribed circle.

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