The Essentials of Geometry (plane)

(j) Prove CF2 - CE2 = AC2 - BC2.

(k) Prove that lines AL, BH, and CM meet at a common point. (Ex. 84, (a).)

(Produce DC to T, making CT = DM, and prove AL, BH, and CM the from the vertices to the opposite sides of ▲ ABT.)

(1) Prove that lines HG, LK, and MC when produced meet at a common point.

(Draw GT and KT, and prove & CGT and CKT rt. ¿.)

85. If BE and CF are medians drawn from vertices B and C of triangle ABC, intersecting at D, prove triangle BCD equivalent to quadrilateral AEDF.

(area BCD area BCF – area BDF.)

86. If D is the middle point of side BC of triangle ABC, E the middle point of AD, F of BE, and G of CF, prove ▲ ABC equivalent to 8 A EFG.

(Draw CE; then, area ABC = 2 area BCE.)

87. If E and Fare the middle points of sides AB and CD, respectively, of parallelogram ABCD, and AF and CE be drawn intersecting BD in H and L, respectively, and BF and DE intersecting AC in K and G, respectively, prove GHKL a parallelogram equivalent to ABCD. (§ 140.)

(If AC and BD intersect at M, AM and DE are medians of ▲ ABD.) 88. Any quadrilateral ABCD is equivalent to a triangle, two of whose sides are equal to diagonals AC and BD, respectively, and include an angle equal to either of the angles between AC and BD. (Produce AC to F, making CF AE; and BD to G, making DG BE. To prove quadrilateral A ABCDEFG. ADFG≈. ΔΑΒC.)

89. If through any point E in diagonal AC of parallelogram ABCD parallels to AD and AB be drawn, meeting AB and CD in F and H, respectively, and BC and AD in G and K, respectively, prove triangles EFG and EHK equivalent.

90. If E is the intersection of diagonals AC and BD of a quadrilateral, and triangles ABE and CDE are equivalent, prove sides AD and BC parallel.

(AABD and ACD are equivalent.)

91. Find the area of a trapezoid whose parallel sides are 28 and 36, and non-parallel sides 15 and 17, respectively.

(By drawing through one vertex of the upper base a || to one of the non-parallel sides, one of the figure may be proved a rt. 4, by Ex. 63, p. 154.)

92. If similar polygons be described upon the legs of a right triangle as homologous sides, the polygon described upon the hypotenuse is equivalent to the sum of the polygons described upon the legs.

(Find, by § 322, the ratio of the area of the polygon described upon each leg to the area of the polygon described upon the hypotenuse.)

93. If E, F, G, and H are the middle points of sides AB, BC, CD, and DA, respectively, of a square, prove that lines AG, BH, CE, and DF form a square equivalent to ABCD.

(First prove ▲ ADG = ^ ABH; then, by § 85, 1, NKL may be proved a rt. Z. By § 131, each side of KLMN may be proved equal to AK. From

ןד

similar ▲ AHK and ADG, AK may be proved equal to 4D)

94. If E is any point in side BC of parallelogram ABCD, and DE be drawn meeting AB produced at F, prove triangles ABE and CEF equivalent.

(AABE + ACDE≈ ^ CDF.)

95. If D is a point in side AB of triangle ABC, find a point E in AC such that triangle ADE shall be equivalent to one-half triangle ABC.

(A DEFACEF)

What restriction is there on the position of D? B

BOOK V.

96. The area of the ring included between two concentric circles is equal to the area of a circle, whose diameter is that chord of the outer circle which is tangent to the inner.

(To prove area of ring = πAC2.)

97. An equilateral polygon circumscribed about a circle is regular if the number of its sides is odd. (§ 345.)

(The polygon can be inscribed in a O.)

98. An equiangular polygon inscribed in a circle is regular if the number of its sides is odd. (§ 345.)

(The polygon can be proved equilateral.)

99. If a circle be circumscribed about a right G triangle, and on each of its legs as a diameter a semicircle be described exterior to the triangle, the sum of the areas of the crescents thus formed is equal to the area of the triangle. (§ 272.)

(To prove area AECG + area BFCH equal to area ABC.)

100. If the radius of the circle is 1, the side, apothem, and diagonal of a regular inscribed pentagon are, respectively,

√(10 − 2 √5), † (1 + √5), and | √(10+2 √5).

(In Fig. of Prop. IX., the apothem of a regular inscribed pentagon is the distance from 0 to the foot of a 1 from B to OA, and its side is twice this. The diagonal is a leg of a rt. A whose hypotenuse is a diameter, and whose other leg is a side of a regular inscribed decagon.)

101. The square of the side of a regular inscribed pentagon, minus the square of the side of a regular inscribed decagon, is equal to the square of the radius. (Ex. 100, and § 359.)

102. The sum of the perpendiculars drawn to the sides of a regular polygon from any point within the figure is equal to the apothem multiplied by the number of sides of the polygon.

(The Is are the altitudes of A which make up the polygon.)

103. In a given equilateral triangle to in

scribe three equal circles, tangent to each other, and each tangent to one, and only one, side of the triangle.

(By § 174, the touch the Is at the same points.)

104. In a given circle to inscribe three equal circles, tangent to each other and to the given circle.