=

(2 ab + a2 + b2 — c2) (2 ab — a2 — b2 + c2)

4 a2

{ (a + b) 2 − c2 } { c2 —-( a − b)2}

4 a2

(a+b+c) (a+b−c) (c+ a−b) (c — a + b).

4 a2

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

2°. To compute the medians of a triangle in terms of the sides.

In ▲ ABC, denoting the values of the sides as in 1o, and the median from 4 by m,

A

since b2 + c2 = 2 m2 + 2({ a)2,

(353)

m = • 1⁄2 √2 (b2 + c2) — a2.

B

D

3°. To compute the bisectors of a triangle in terms of the sides.

In ▲ ABC, denoting the values of the sides as in 1o, and the bisector from 4, by d (see diagram for Prop. XVI.);

Substituting these values in 1°, and simplifying,

b + c

4°. To compute the radius of the circumscribed circle in terms of the sides of a triangle ABC.

In ▲ ABC, denoting the values of the sides as in 1o, and the radius by R (see diagram for Prop. XV.);

5°. To compute the area of a triangle in terms of its sides.

Denoting the values of the sides as in 1o, that of the altitude by h, and of the area by S,

2

since Sah (331), and h == √s(s-a) (s-b) (s-c), (1°)

a

s = √s( s − a) (s — b) ( s − c).

SCHOLIUM. If the triangle is equilateral, i.e., if a=b=c,

6°. To compute the area of a triangle in terms of the sides and the radius of the circumscribing circle.

Geom. - 13

Denoting the values of the sides, altitude, and area as

in 5o,

505. How many different altitudes can each of the following figures have An equilateral triangle? An isosceles triangle? A scalene triangle? A square? A rectangle? A trapezoid? A trapezium?

506. A side of an equilateral triangle is 6.* What is its altitude? 507. An arm and the base of an isosceles triangle are 18 and 16 respectively. What are its altitudes ?

508. The area of a triangle is 180; its sides are 30, 60, and 40, respectively. What are its altitudes ?

509. The area of a triangle is 252; its altitudes are 8, 12, and 14, respectively. What are its sides?

510. The sides of a rectangle are 65 and 32 respectively. What are its area, perimeter, and diagonal?

511. The altitude and base of a triangle being 23 and 10 respectively, what is its area?

512. The area of a triangle is 221 sq. ft.; its base is 53 yds. What is its altitude in inches?

513. The bases of two parallelograms are 15 and 16 respectively; their altitudes are 8 and 10 respectively. What is the ratio of their areas?

514. Two triangles of equal areas have their bases 26 in. and 3 ft. respectively. What is the ratio of their altitudes?

* Remember that the given abstract numbers are numerical measures.

515. The bases of a trapezoid are 23 in. and 17 in. respectively, its altitude being 21 ft. What is its area?

516. In ▲ ABC, AB = 42, AC = 34. If DE cut off AD = 30 and AE = 15, what is the ratio of AABC to ▲ ADE?

517. What should be the length of a ladder such that, having its foot 15 ft. from the wall, it may reach a window 20 ft. from the ground?

B

D

E

518. Two chords intersect so that the segments of one are 12 and 7 respectively. If a segment of the other is 10, what is its second segment?

519. What are the altitudes of a triangle whose sides are 12, 15, 9, respectively?

520. What are the medians of the same triangle ?

521. What are the bisectors of the angles of the same triangle?

522. What is the radius of the circle circumscribing the same triangle ?

523. What is the area of the same triangle ?

524. What is the area of a triangle whose sides are 6, 5, 5, respectively?

525. What is the area of an equilateral triangle whose side is 3?

526. What is the area of an equilateral triangle whose altitude is 11 ?

527. The sides of a right triangle are 25, 24, 7. medians and its altitude upon the hypotenuse?

What are its

528. From the same point a tangent and a secant being drawn, if the secant and its external segment are as 27 to 3, what is the length of the tangent?

529. If from the point just referred to a second secant be drawn, so that its external segment is 8, what will be the length of the secant?

530. Two secants drawn from the same point have external segments of 5 and 3 respectively. If the first secant is 27, what are the internal segments?

PROBLEMS.

531. Construct an isosceles triangle on the same base as a given triangle, and equivalent to it.

532. Construct a right isosceles triangle equivalent to a given

square.

533. Construct a parallelogram having a given angle upon the same base as a given square, and equivalent to it.

534. Divide a given line into two segments such that their squares shall be as 7 is to 5.

535. Bisect a given parallelogram (1) by a line passing through a given point; (2) by a perpendicular to a side; (3) by a line parallel to a given line.

536. Bisect a given triangle by a line drawn through a given point P in one of the sides.

537. Cut off one nth of a triangle by a line drawn through a given point in one of the sides.

538. Bisect a quadrilateral by a line drawn through one of the vertices.

539. Cut off from a quadrilateral one nth

part by a line drawn through one of the vertices.

540. Bisect a triangle by a line parallel to the base.

541. Bisect a triangle by a line perpendicular to the base.

542. Find a point within a triangle such that lines joining the point with the vertices shall divide the triangle into three equivalent triangles.

543. Bisect a trapezoid by a line drawn parallel to the bases.

544. Bisect a trapezoid by a line drawn through a given point in one of the bases.

545. Construct a triangle equivalent to a given triangle, and having one side equal to a given line.

546. Construct a right triangle equivalent to a given triangle, and having one arm of a given length.