Ex. 543. The perpendiculars from two vertices of a triangle upon the opposite sides divide each other into segments reciprocally proportional.

Ex. 544. The perpendicular from any point of a circumference upon a chord is the mean proportional between the perpendiculars from the same point upon the tangents drawn at the extremities of the chord.

Ex. 545. In an isosceles right triangle either leg is the mean proportional between the hypotenuse and the perpendicular upon it from the vertex of the right angle.

Ex. 546. If two circles intersect in the points A and B, and through A any secant CAD is drawn limited by the circumferences at C and D, the straight lines BC, BD are to each other as the diameters of the circles.

Ex. 547. The area of a triangle is equal to half the product of its perimeter by the radius of the inscribed circle.

Ex. 548. The perimeter of a triangle is to one side as the perpendicular from the opposite vertex is to the radius of the inscribed circle.

Ex. 549. If three straight lines AA', BB', CC', drawn from the vertices of a triangle ABC to the opposite sides, pass through a common point within the triangle, then

Ex. 550. ABC is a triangle, M the middle point of AB, P any point in AB between A and M. If MD is drawn parallel to PC, meeting BC at D, the triangle BPD is equivalent to half the triangle ABC.

Ex. 551. Two diagonals of a regular pentagon, not drawn from a common vertex, divide each other in extreme and mean ratio.

Ex. 552. If all the diagonals of a regular pentagon are drawn, another regular pentagon is thereby formed.

Ex. 553. The area of an inscribed regular dodecagon is equal to three times the square of the radius.

Ex. 554. The area of a square inscribed in a semicircle is equal to two tifths the area of the square inscribed in the circle.

Ex. 555. The area of a circle is greater than the area of any polygon of equal perimeter.

Ex. 556. The circumference of a circle is less than the perimeter of any polygon of equal area.

PROBLEMS OF LOCI.

Ex. 557. Find the locus of the centre of the circle inscribed in a triangle that has a given base and a given angle at the vertex.

Ex. 558. Find the locus of the intersection of the altitudes of a triangle that has a given base and a given angle at the vertex.

Ex. 559. Find the locus of the extremity of a tangent to a given circle, if the length of the tangent is equal to a given line.

Ex. 560. Find the locus of a point, tangents drawn from which to a given circle form a given angle.

Ex. 561. Find the locus of the middle point of a line drawn from a given point to a given straight line.

Ex. 562. Find the locus of the vertex of a triangle that has a given base and a given altitude.

Ex. 563. Find the locus of a point the sum of whose distances from two given parallel lines is equal to a given length.

Ex. 564. Find the locus of a point the difference of whose distances from two given parallel lines is equal to a given length.

Ex. 565. Find the locus of a point the sum of whose distances from two given intersecting lines is equal to a given length.

Ex. 566. Find the locus of a point the difference of whose distances from two given intersecting lines is equal to a given length.

Ex. 567. Find the locus of a point whose distances from two given points are in the given ratio m: n.

Ex. 568. Find the locus of a point whose distances from two given parallel lines are in the given ratio m: n.

Ex. 569. Find the locus of a point whose distances from two given intersecting lines are in the given ratio m: n.

Ex. 570. Find the locus of a point the sum of the squares of whose distances from two given points is constant.

Ex. 571. Find the locus of a point the difference of the squares of whose distances from two given points is constant.

Ex. 572. Find the locus of the vertex of a triangle that has a given base and the other two sides in the given ratio m: n.

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 twc 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. 597. 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