Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/395

 343. Perpendiculars are drawn from any point within an equilateral triangle on the three sides: shew that their sum is invariable.

344. Find a point within a triangle such that if straight lines be drawn from it to the three angular points the triangle will be divided into three equal triangles.

345. From a point E in the common base of two triangles ACB, ADB, straight lines are drawn parallel to AC, AD, meeting BC, BD at F, G: shew that FG is parallel to CD.

346. From any point in the base of a triangle straight lines are drawn parallel to the sides: shew that the intersection of the diagonals of every parallelogram so formed lies in a certain straight line.

347. In a triangle ABC a straight line AD is drawn perpendicular to the straight line BD which bisects the angle B: shew that a straight line drawn from D parallel to BC will bisect AC.

348. ABC is a triangle; any straight line parallel to BC meets AB at D and AC at E; join BE and CD meeting at F: shew that the triangle ADF is equal to the triangle AEF.

349. ABC is a triangle; any straight line parallel to BC meets AB at D and AC at E; join BE and CD meeting at F: shew that if AF be produced it will bisect BC.

350. If two sides of a quadrilateral figure be parallel to each other, any straight line drawn parallel to them will cut the other sides, or those sides produced, proportionally.

351. ABC is a triangle; it is required to draw from a given point P, in the side AB, or AB produced, a straight line to AC or AC produced, so that it may be bisected by BC.

VI. 3, A.

352. The side BC of a triangle ABC is bisected at D. and the angles ADB, ADC are bisected by the straight, lines DE, DF, meeting AB, AC at E,F respectively: shew that EF is parallel to BC.

353. AB is a diameter of a circle, CD is a chord at right angles to it, and E is any point in CD; AE and BE