THEOREMS 248. The locus of a point equidistant from the ends of a given line is the perpendicular-bisector of that line. 249. The locus of a point at a given distance from a given point is the circumference described from the point with the given distance as radius. 250. The locus of a point that is at a given distance from a given straight line consists of two lines parallel to the given line at the given distance. 251. The locus of a point equidistant from two given parallel lines is a third parallel, bisecting any line ending in the given parallels. 252. The locus of a point equidistant from two intersecting straight lines, consists of the bisectors of the included angles. Ex. 429. Find the locus of the vertex of all right angles whose sides pass through two given points. Ex. 430. Find the locus of the midpoints of the radii of a given circle. EXERCISES In the following exercises, state under what conditions no point, one point, or several points may be found. Ex. 431. In a given line, AB, find a point at a given distance, d, from a given point, C. Ex. 422. In a given line, AB, find a point at a given distance, d, from a given line, CD. Ex. 433. In a given line, AB, find a point equidistant from two given points, P and Q. Ex. 434. In a given circumference, find a point at a given distance, d, from a given point, C. Ex. 435. In a given circumference, find a point equidistant from two given parallel lines, CD and EF. Ex. 436. In a given circumference, find a point equidistant from two given intersecting lines, CD and EF. Ex. 437. Find a point equidistant from two given intersecting lines, AB and CD, and at a given distance from a given point, E. Ex. 438. Find a point equidistant from two given intersecting lines, AB and CD, and at a given distance from a given line, EF. Ex. 439. Find a point equidistant from two given intersecting lines, AB and CD, and equidistant from two given points, E and F. Ex. 440. Find a point equidistant from two given points, and having a given distance from a given point, E. Ex. 441. Find a point equidistant from two given points and equidistant from two given parallel lines, EF and GH. Ex. 442. Find a point equidistant from two given parallel lines and equidistant from two given intersecting lines, EF and GH. Ex. 443. Find a point at a given distance, d, from a given line, AB, and equidistant from two given points, E and F. Ex. 444. Find a point having a given distance, d, from a given line, AB, and equidistant from two given parallel lines, EF and GH. Ex. 445. Find a point having a given distance, d, from a given line, AB, and equidistant from two given intersecting lines, EF and GH. Ex. 446. Find the locus of the center of a circle that passes through two given points. Ex. 447. Find the locus of the center of a circle that touches two given lines. Ex. 448. Find the locus of the center of a circle which has a given radius and touches a given line. Ex. 449. Find the locus of the center of a circle which has a given radius and touches a given circle. Ex. 450. Find the locus of the center of a circle touching a given line at a given point. Ex. 451. Find the locus of the center of a circle that touches a given circle in a given point. To construct a circle having a given radius: Ex. 452. Touching a given line and passing through a given point Ex. 453. Touching two given lines. Ex. 454. Passing through a point and touching a given circle. Ex. 455. Touching two given circles. Ex. 456. Touching a given circle and a given line. 253. No general method can be given for the solution of exercises; a great many, however, can be solved (1) By a gradual putting together of the given parts. (2) By means of an analysis. (3) By means of loci. MISCELLANEOUS EXERCISES (Remark 233.) Ex. 457. Through a given point, to draw a line cutting off equal parts on the sides of a given angle. Ex. 458. Through a given point, to draw a line making a given angle with a given line. Construct an isosceles triangle, having given: Ex. 459. The base and the altitude upon an arm. Ex. 460. The altitude upon the base and the vertical angle. Ex. 461. The vertical angle and the sum of one arm and the base. Ex. 462. The perimeter and the base angles. Construct a right triangle, having given : Ex. 463. One acute angle and the altitude upon the hypotenuse. Ex. 464. The altitude upon the hypotenuse and one of the segments of the hypotenuse. Ex. 465. The sum of the arms and one acute angle. Ex. 466. To find a point in one side of a triangle which is equidistant from the other two sides. Ex. 467. Find the locus of the vertex of a right triangle, having a given hypotenuse. Ex. 468. From a point P, in the circumference of a circle, to draw a chord, having a given distance from the center. Ex. 469. In a given circle, to draw a diameter having a given distance from a given point. Ex. 470. Through a given point without a circle, to draw a secant having a given distance from the center. Ex. 471. Through two given points in a circumference, to draw two equal parallel chords. Ex. 472. Trisect a given straight angle. Ex. 473. Trisect a given right angle. Ex. 474. Through a given point, to draw a line of given length terminating in two given parallel lines. Ex. 475. Through a given point, to draw a line making equal angles with two given lines. * Ex. 476. To bisect an angle formed by two lines, without producing them to their intersection. Ex. 487. The difference between the diagonal and the side. Ex. 488. The sum of the diagonal and the side. To construct a rectangle, having given: Ex. 489. One side and the diagonal. Ex. 490. The angle formed by the diagonals and one side. Ex. 491. The perimeter and the diagonal. To construct a rhombus, having given: Ex. 492. The two diagonals. Ex. 493. The perimeter and one diagonal. Ex. 495. The altitude and the base. Ex. 496. The altitude and one angle. To construct a parallelogram, having given : Ex. 497. Two sides and one altitude. Ex. 498. Two sides and an angle. Ex. 499. One side and the two diagonals. Ex. 500. One side, one angle, and one diagonal. Ex. 501. The diagonals and the angle formed by the diagonals. 254. In the analysis of a problem relating to a trapezoid, draw a line through one vertex, A, either parallel to the opposite arm, DC, or parallel to a diagonal, DB. B To construct a trapezoid, having given : Ex. 502. The four sides. Ex. 503. The bases and the base angles. Ex. 504. The bases, another side, and one base angle. Ex. 505. The bases and the diagonals. Ex. 506. One base, the diagonals, and the angle formed by the diagonals. Ex. 507. To draw a common exterior tangent to two given circles. Ex. 508. To draw a common interior tangent to two given circles. Ex. 509. About a given circle, to circumscribe a triangle, having given the angles. Ex. 510. Find the locus of the midpoints of the chords that pass through a given point in the circumference. Ex. 511. Find the locus of the midpoints of the secants that pass through a given point without a circle. Ex. 512. In a given circle, to inscribe a triangle, having given the angles. *Ex. 513. From a given point in a circumference, to draw a chord that is bisected by a given chord. |