415. Chords drawn from the point of contact of a tangent have their segments made by any chord parallel to the tangent, inversely proportional. 416. If a tangent is intercepted between two parallel tangents to the same circle, its segments made by the point of contact have the radius as mean proportional. 417. If three circles intersect each other, their three common chords pass through the same point. 418. If through the mid point of a side of a triangle a line be drawn intersecting a second side, the third side produced, and a line parallel to the first through the opposite vertex, the line will be divided harmonically. PROBLEMS. 419. To a given circle draw a tangent that shall be perpendicular to a given line. 420. To a given circle draw a tangent that shall be parallel to a given line. 421. To a given circle draw two tangents including a given angle. 422. In a given straight line, find a point such that the tangents drawn from it to a given circle shall include the greatest angle. 423. In a chord produced, find a point such that the tangent from that point shall be equal to a given line. 424. From a given center describe a circle tangent to a given circle. 425. Describe a circumference passing through a given point and touching a given circle in a given point. 426. Describe two circles with given radii, so intersecting that their common chord shall have a given length not greater than the lesser diameter. 427. With a given radius, describe a circle tangent to two given circles. 428. Draw a common exterior tangent to two given circles. 429. Draw a common interior tangent to two given circles. 430. Find a point such that the tangents drawn from it to the outer sides of two tangent circles shall include a given angle. 431. Divide any side of a triangle into two parts proportional to the other sides. 432. Divide any side of a triangle into three parts proportional to the three sides. 433. From a given line cut off a part that shall be a mean proportional between the remainder and another given line. 434. Through a given point within a circle, draw a chord there divided in the same ratio as a given chord through that point. 435. From a given point without a circle draw a secant divided by the circumference in a given ratio. 436. From a given point in a given arc draw a chord bisected by the chord of the given arc. 437. In a given circle place a chord that shall be trisected by two given radii at right angles to each other. 438. In a given circle place a chord parallel to a given chord, and having to it a given ratio not greater than that of the diameter to the given chord. 439. Through one of the points of intersection of two given circles draw a secant forming chords that are in a given ratio. 440. Inscribe a square in a given triangle. 441. Inscribe a square in a given segment of a circle. 442. In a given semicircle inscribe a rectangle similar to a given rectangle. 443. In a given circle inscribe a triangle similar to a given triangle. 444. About a given circle circumscribe a triangle similar to a given triangle. 445. In a given triangle construct a parallelogram similar to a given parallelogram. 446. Construct a triangle having given the base, the vertical angle, and the length of the bisector of that angle. LOCI. 447. Find the locus of the center of each circumference that passes through two given points. 448. Find the locus of the center of each circle that is tangent to a given circle at a given point. 449. Find the locus of the center of each circle of given radius that is tangent to a given circle. 450. Find the locus of the center of each circle that is tangent to a given line at a given point. 451. Find the locus of the center of each circle that is tangent to two given intersecting lines. 452. Find the locus of the points from which pairs of tangents of a given length may be drawn to a given circle. 453. Find the locus of the mid point of any chord that passes through a given point in a given circle. 454. Find the locus of the mid point of any secant that can be drawn from a given point to a given circumference. 455. Find the locus of the vertex of any triangle constructed on a given base, with a given vertical angle. 456. Find the locus of a point whose distances from two given points are in a given ratio. 457. Find the locus of a point whose distances from two given straight lines are in a given ratio. 458. Find the locus of a point the sum of whose distances from two given straight lines is equal to a given line. 459. Find the locus of a point the difference of whose distances from two given straight lines is equal to a given line. 460. Find the locus of the points that divide the chords of a given circle so that the rectangle of their segments is equal to a given square. BOOK V. AREAS AND THEIR COMPARISON. QUADRILATERALS. 313. The area of a plane figure is the quantity of its surface as measured by the unit of surface, or is the numerical measure of that quantity. 314. Figures that are not similar but have equal areas are said to be equivalent. 315. The base of a polygon is any side on which we choose to regard it as constructed. 316. The altitude of a polygon is the perpendicular distance to the base from the remotest vertex or from a side parallel to the base or the base produced. A O H AP B A P B Thus in each of the figures above, the perpendicular CP is the altitude of the figure when AB is taken as base. It is obvious that two triangles having their bases in the same line and the opposite vertex common, as ACB and ACD, have the same altitude. PROPOSITION I. THEOREM. 317. Rectangles with equal altitudes are to each other as their bases. Given Two rectangles AC, A'C', with equal altitudes AB, A'B', and bases BC, B'c'; To Prove: Rectangle AC: rectangle A'c'= BC : B'C'. 1°. When BC and B'C' are commensurable. Let BE be a common measure of BC and B'C', so that BE can be laid off 5 times on B'C' and 8 times on BC. Through each point of division draw perpendiculars to the opposite side of the rectangle. The figures thus formed are equal rectangles. (144) Since BC and B'C' contain 8 and 5 parts, respectively, each equal to BE, (Const.) AC and A'C', resp., contain 8 and 5 parts, each equal to AE. .. BC: B'C' 8: 5, and AC: A'c'= 8:5; (225) .. rect. AC rect. A'C' BC: B'C'. Q.E.D. (232""') = 2o. When BC and B'C' are incommensurable. Suppose B'C' divided into any number of equal parts n, and that BC contains this nth part of B'C' m times with a remainder EC. Draw EF perpendicular to BC. Since BE and B'C' are commensurable, (Const.) BE m rect. AE = = B'C' n rect. A'C' (1°) |