6. The straight lines drawn at right angles to the sides of a triangle from their mid-points, meet in a point which is equally distant from the three vertices of the triangle. B E (A Standard Theorem.) Let D, E, F be the mid-points of BC, CA, AB. Use Euc. I. 8 to prove ▲ BOD=▲ COD, Observe that the point O need not be within the triangle. DEFINITION.-Three or more straight lines which meet in a point are said to be concurrent. Thus Ex. 6 shows that the straight lines drawn at right angles to the sides of a triangle from their mid-points are concurrent. 7. If the opposite sides of a quadrilateral be equal, the opposite angles shall also be equal. Use Euc. I. 8. 8. The straight line which joins the vertices of two isosceles triangles on the same base bisects both vertical angles. 9. ABCD is a quadrilateral in which the side AB-CD, and the diagonal AC=BD. Show that A= 2 D, < B= < C, and that, if AC and BD intersect in O, the As OAD, OBC are isosceles. § 4. (Bookwork, EUCLID, I. 1-14.) 1. The bisectors of the adjacent angles, formed by one straight line standing on another straight line, are at right angles to each other. E F B (A Standard Theorem.) Let AB stand on CD and let BE, BF bisect the angles ABC, ABD; it is required to prove that DEFINITIONS.-Two angles which together make up a right angle are said to be complementary angles, and either angle is called the complement of the other. Two angles which together make up two right angles are said to be supplementary angles, and either angle is called the supplement of the other. Thus in Ex. 1, the 4s ABE, ABF are complementary, and the SABC, ABD are supplementary. 2. The bisectors of the four angles, which two intersecting straight lines make with each other, form two straight lines which are perpendicular to each other. Let AB, CD be the given straight lines, etc. Show as in Ex. 1 that s POQ, QOR, etc., are right angles, and apply Euc. I. 14. 3. ABC is a triangle in which AB > AC. The bisector of the A meets the base in K. Show that AKB is obtuse. From AB (the greater) cut off AD=AC. Use Euc. I. 4 to show that LAKD LAKC. Deduce = AKB> AKC, and use Euc. I. 13. 4. If from a point within a right angle perpendiculars be drawn to the lines containing the angle, and each perpendicular be produced its own length, the extremities of the produced lines and the vertex of the right angle shall be in one straight line. DEFINITION.-Three or more points which are in the same straight line are said to be collinear. Thus, in Ex. 4, M, B, N are shown to be collinear. 5. Four straight lines, OA, OB, OC, OD, meet in a point making < AOD= < BOC, and AOB= COD. Prove that AO, OC (and BO, OD) are in the same straight line. Use Euc. I. 13, Cor. ii.,* to shows AOB, BOC supplementary. * This corollary is sometimes given as a corollary to Euc. I. 15. 6. P is a point within the ▲ ABC. Show that of the 4s BPC, CPA, APB, at least two must be obtuse. Use Euc. I. 13, Cor. ii. 7. Prove § 2, Ex. 2, for the case in which P, Q, R are on the sides produced. 8. The corner of the leaf of a book is folded down. Show that the bisector of the angles formed by the edges of the leaf and the edges of the folded part are at right angles to the crease. § 5. (Bookwork, EUCLID, I. 1-20.) 1. The straight line drawn from any angular point of a triangle to the mid-point of the opposite side is less than half the sum of the other two sides. DEFINITION. ... AC+AB>2AD, or AD< (AB+AC). The straight line drawn from any vertex of a triangle to the mid-point of the opposite side is called the Median from that vertex. Thus Ex. 1 shows that the median drawn from any vertex is less than half the sum of the sides containing the angle. 2. In an isosceles triangle the medians drawn from the extremities of the base are equal. |