PROBLEMS OF CONSTRUCTION.

Ex. 573. To divide a given trapezoid into two equivalent parts by a line parallel to the bases.

Ex. 574. To divide a given trapezoid into two equivalent parts by a line through a given point in one of the bases.

Ex. 575, To construct a regular pentagon, given one of the diagonals.

Ex. 576. To divide a given straight line into two segments such that their product shall be the maximum.

Ex. 577. To find a point in a semicircumference such that the sum of its distances from the extremities of the diameter shall be the maximum.

Ex. 578. To draw a common secant to two given circles exterior to each other such that the intercepted chords shall have the given lengths a, b.

Ex. 579. To draw through one of the points of intersection of two intersecting circles a common secant which shall have a given length.

Ex. 580. To construct an isosceles triangle, given the altitude and one of the equal base angles.

Ex. 581. To construct an equilateral triangle, given the altitude.

Ex. 582. To construct a right triangle, given the radius of the inscribed circle and the difference of the acute angles.

Ex. 583. To construct an equilateral triangle so that its vertices shall lie in three given parallel lines.

Ex. 584. To draw a line from a given point to a given straight line which shall be to the perpendicular from the given point as m : n.

Ex. 585. To find a point within a given triangle such that the perpendiculars from the point to the three sides shall be as the numbers m, n, p.

Ex. 586. To draw a straight line equidistant from three given points.

Ex. 587. To draw a tangent to a given circle such that the segment intercepted between the point of contact and a given straight line shall have a given length.

Ex. 588. To inscribe a straight line of a given length between two given circumferences and parallel to a given straight line.

Ex. 589. To draw through a given point a straight line so that its distances from two other given points shall be in a given ratio.

Ex. 590. To construct a square equivalent to the sum of a given triangle and a given parallelogram.

Ex. 591. To construct a rectangle having the difference of its base and altitude equal to a given line, and its area equivalent to the sum of a given triangle and a given pentagon.

Ex. 592. To construct a pentagon similar to a given pentagon and equivalent to a given trapezoid.

Ex. 593. To find a point whose distances from three given straight lines shall be as the numbers m, n, p.

Ex. 594. Given an angle and two points P and P' between the sides of the angle. To find the shortest path from P to P' that shall touch both sides of the angle.

Ex. 595. To construct a triangle, given its angles and its area.

Ex. 596. To transform a given triangle into a triangle similar to another given triangle.

Ex. 97. Given three points A, B, C. To find a fourth point P such that the areas of the triangles APB, APC, BPC shall be equal.

Ex. 598. To construct a triangle, given its base, the ratio of the other sides, and the angle included by them.

Ex. 599. To divide a given circle into n equivalent parts by concentric circumferences.

Ex. 600. In a given equilateral triangle to inscribe three equal circles tangent to each other, each circle tangent to two sides of the triangle.

Ex. 601. Given an angle and a point P between the sides of the angle. To draw through P a straight line that shall form with the sides of the angle a triangle with the perimeter equal to a given length a.

Ex. 602. In a given square to inscribe four equal circles, so that each circle shall be tangent to two of the others and also tangent to two sides of the square.

Ex. 603. In a given square to inscribe four equal circles, so that each circle shall be tangent to two of the others and also tangent to one side of the square.

SOLID GEOMETRY.

BOOK VI.

LINES AND PLANES IN SPACE.

DEFINITIONS.

492. DEF. A plane is a surface such that a straight line joining any two points in it lies wholly in the surface. A plane is understood to be indefinite in extent; but is usually represented by a parallelogram lying in the plane.

493. DEF. A plane is said to be determined by given lines or points, if no other plane can contain the given lines or points without coinciding with that plane.

495. COR. 2. A straight line and a point not in the line determine a plane.

For, if a plane containing a straight line AB and any point C not in AB is made to revolve either way about AB, it will no longer contain the point C.

496. COR. 3. Three points not in a straight line determine a plane.

For by joining two of the points we have a straight line and a point without it, and these determine the plane. § 495

497. COR. 4. Two intersecting lines determine a plane. For the plane containing one of these lines and any point. of the other line not the point of intersection is determined.

§ 495

M

B

N

498. COR. 5. Two parallel lines determine a plane. For two parallel lines lie in a plane (§ 103), and a plane containing either parallel and a point in the other is determined.

499. DEF. When we suppose a plane to be drawn through given points or lines, we are said to pass the plane through the given points or lines.

500. DEF. When a straight line is drawn from a point to a plane, its intersection with the plane is called its foot.

501. DEF. A straight line is perpendicular to a plane, if it is perpendicular to every straight line drawn through its foot in the plane; and the plane is perpendicular to the line.

502. DEF. A straight line and a plane are parallel if they cannot meet, however far both are produced.

503. DEF. A straight line neither perpendicular nor parallel to a plane is said to be oblique to the plane.

504. DEF. Two planes are parallel if they cannot meet, however far they are produced.

505. DEF. The intersection of two planes contains all the points common to the two planes.

LINES AND PLANES.

PROPOSITION I. THEOREM.

506. If two planes cut each other, their intersection is a straight line.

Let MN and PQ be two planes which cut one another.

To prove that their intersection is a straight line.
Proof. Let A and B be two points common to the two planes.
Draw a straight line through the points A and B.

Then the straight line AB lies in both planes. § 492

No point not in the line AB can be in both planes; for one plane, and only one, can contain a straight line and a point without the line.

§ 495 Therefore, the straight line through A and B contains all the points common to the two planes, and is consequently the intersection of the planes.

§ 505

Q. E. D.