For let a plane XY intersect all the arms of the plane angles on the same side of the vertex at the points A, B, C, D, E: and let AB, BC, CD, DE, EA be the common sections of the plane XY with the planes of the several angles. Within the polygon ABCDE take any point o; and join O to each of the vertices of the polygon. Then since the _8 SAE, SAB, EAB form the trihedral angle A, .. the <* SAE, SAB are together greater than the 2 EAB; XI. 20. that is, the Ľ SAE, SAB are together greater than the < OAE, OAB. Similarly, the 4* SBA, SBC are together greater than the 2* OBA, OBC: and so on, for each of the angular points of the polygon. Thus by addition, the sum of the base angles of the triangles whose vertices are at s, is greater than the sum of the base angles of the triangles whose vertices are at O. But these two systems of triangles are equal in number; .:. the sum of all the angles of the one system is equal to the sum of all the angles of the other. It follows that the sum of the vertical angles at S is less than the sum of the vertical angles at O. But the sum of the angles at O is four rt. angles ; .. the sum of the angles at S is less than four rt. angles. Q.E.D. NOTE. This proposition was not given in this form by Euclid, who established its truth only in the case of trihedral angles. The above demonstration, however, applies to all cases in which the polygon ABCDE is convex, but it must be observed that without this condition the proposition is not necessarily true. A solid angle is convex when it lies entirely on one side of each of the infinite planes which pass through its plane angles. If this is the case, the polygon ABCDE will have no re-entrant angle. And it is clear that it would not be possible to apply xi. 20 to a vertex at which a re-entrant angle existed. EXERCISES ON BOOK XI. a 1. Equal straight lines drawn to a plane from a point without it have equal projections on that plane. 2. If S is the centre of the circle circumscribed about the triangle ABC, and if SP is drawn perpendicular to the plane of the triangle, shew that any point in SP is equidistant from the vertices of the triangle. 3. Find the locus of points in space equidistant from three given points. 4. From Example 2 deduce a practical method of drawing a perpendicular from a given point to a plane, having given ruler, compasses, and a straight rod longer than the required perpendicular. 5. Give a geometrical construction for drawing a straight line equally inclined to three straight lines which meet in a point, but are not in the same plane. 6. In a gauche quadrilateral (that is, a quadrilateral whose sides are not in the same plane) if the middle points of adjacent sides are joined, the figure thus formed is a parallelogram. 7. AB and AC are two straight lines intersecting at right angles, and from B a perpendicular BD is drawn to the plane in which they are: shew that AD is perpendicular to AC. 8. If two intersecting planes are cut by two parallel planes, the lines of section of the first pair with each of the second pair contain equal angles. 9. If a straight line is parallel to a plane, shew that any plane passing through the given straight line will intersect the given plane in a line of section which is parallel to the given line. 10. Two intersecting planes pass one through each of two parallel straight lines; shew that the common section of the planes is parallel to the given lines. 11. If a straight line is parallel to each of two intersecting planes, it is also parallel to the common section of the planes. 12. Through a given point in space draw a straight line to intersect each of two given straight lines which are not in the same plane. 13. If AB, BC, CD are straight lines not all in one plane, shew that a plane which passes through the middle point of each one of them is parallel both to AC and BD. 14. From a given point A a perpendicular AB is drawn to a plane XY; and a second perpendicular AE is drawn to a straight line CD in the plane XY: shew that EB is perpendicular to CD. 15. From a point A two perpendiculars AP, AQ are drawn one to each of two intersecting planes: shew that the common section of these planes is perpendicular to the plane of AP, AQ. 16. From A, a point in one of two given intersecting planes, AP is drawn perpendicular to the first plane, and AQ perpendicular to the second : if these perpendiculars meet the second plane at P and Q, shew that PQ is perpendicular to the common section of the two planes. 17. A, B, C, D are four points not in one plane, shew that the four angles of the gauche quadrilateral ABCD (see Ex. 6, p. 444) are together less than four right angles. 18. OA, OB, OC are three straight lines drawn from a given point O not in the same plane, and OX is another straight line within the solid angle formed by OA, OB, OC: shew that (i) the sum of the angles AOX, BOX, COX is greater than half the sum of the angles AOB, BOC, COA. (ii) the sum of the angles AOX, COX is less than the sum of the angles AOB, COB. (iii) the sum of the angles AOX, BOX, COX is less than the sum of the angles AOB, BOC, COA. 19. OA, OB, OC are three straight lines forming a solid angle at O, and OX bisects the plane angle AOB ; shew that the angle XOC is less than half the sum of the angles AOC, BOC. 20. If a point is equidistant from the angles of a right-angled triangle and not in the plane of the triangle, the line joining it with the middle point of the hypotenuse is perpendicular to the plane of the triangle. 21. The angle which a straight line makes with its projection on a plane is less than that which it makes with any other straight line which meets it in that plane. 22. Find a point in a given plane .such that the sum of its distances from two given points (not in the plane but on the same side of it) may be a minimum. 23. If two straight lines in one plane are equally inclined to another plane, they will be equally inclined to the common section of these planes. 24. PA, PB, PC are three concurrent straight lines, each of which is at right angles to the other two: PX, PY, PZ are perpendiculars drawn from P to BC, CA, AB respectively. Shew that XYZ is the pedal triangle of the triangle ABC. 25. PA, PB, PC are three concurrent straight lines, each of which is at right angles to the other two, and from P a perpendicular PO is drawn to the plane of ABC: shew that 0 is the orthocentre of the triangle ABC. THEOREMS AND EXAMPLES ON BOOK XI. DEFINITIONS. (i) Lines which are drawn on a plane, or through which a plane may be made to pass, are said to be co-planar. (ii) The projection of a line on a plane is the locus of the feet of perpendiculars drawn from all points in the given line to the plane. a THEOREM 1. a straight line. The projection of a straight line on a plane is itself B Х Let AB be the given st. line, and XY the given plane. It is required to shew that the locus of p is a st. line. XI. 6. XI. 7. .. the point p is in the common section of the planes Ab, XY; that is, p is in the st. line ab. But p is any point in the projection of AB, :: the projection of AB is the st. line ab. Q.E.D. THEOREM 2. Draw a perpendicular to each of two straight lines which are not in the same plane. Prove that this perpendicular is the shortest distance between the two lines. XI. 9. Let AB and CD be the two straight lines, not in the same plane. (i) It is required to draw a st. line perp. to each of them. Through E, any point in AB, draw EF parł to CD. Let XY be the plane which passes through AB, EF. From H, any point in CD, draw HK perp. to the plane XY. xi. 11. And through K, draw KQ par to EF, cutting AB at Q. Then KQ is also parl to CD; and CD, HK, KQ are in one plane. XI. 7. From Q, draw QP par to HK to meet CD at P. Then shall PQ be perp. to both AB and CD. For, since HK is perp. to the plane XY, and PQ is par to HK, Constr. :. PQ is perp. to the plane XY; XI. 8. :: PQ is perp. to AB, which meets it in that plane. XI. Def. 1. For a similar reason PQ is perp. to QK, :: PQ is also perp. to CD, which is parl to QK. (ii) It is required to shew that PQ is the least of all st. lines drawn from AB to CD. Take He, any other st. line drawn from AB to CD. Ex. 1, p. 429. :: HE is also greater than PQ. Q.E.D. perp. HK. |