Plane and solid geometry . %% t r ^^ ^^^^^ ^^ / Given line BQ. To construct an equilateral triangle on BG. I. Construction 1. With B as center and BO as radius, construct circle CAD. 2. With C as center and BC as radius, construct circle BMA. 3. Connect point Ay at which the circumferences intersect,with B and C. 4. A ABC is the required triangle. IL Proof Aboumbnt 1. ABsbBC ajid CA = BO. 2. .\AB = BC^OA. 3. /. A .4^(7 is equilateral Reasons L All radii of the same circle are equal. § Things equal to the same thing are equal to each other. § 54, A A having its three sides equ
Plane and solid geometry . %% t r ^^ ^^^^^ ^^ / Given line BQ. To construct an equilateral triangle on BG. I. Construction 1. With B as center and BO as radius, construct circle CAD. 2. With C as center and BC as radius, construct circle BMA. 3. Connect point Ay at which the circumferences intersect,with B and C. 4. A ABC is the required triangle. IL Proof Aboumbnt 1. ABsbBC ajid CA = BO. 2. .\AB = BC^OA. 3. /. A .4^(7 is equilateral Reasons L All radii of the same circle are equal. § Things equal to the same thing are equal to each other. § 54, A A having its three sides equal is equilateral. § 95. IIL Discussion This construction is always possible, and there is only onesolution. (See § 116.) 38 PLANE GEOMETRY Propositiox VII. Problem 125. With a given vertex and a given side, to constructan angle equal to a given Given vertex A, side AB, and Z CDE. To construct an Z equal to Z CDE and having A as vertexand AB as side. I. Construction 1. With D as center, and with any convenient radius, de-scribe an arc intersecting the sides of Z D at i^ and G, respec-tively. 2. With A as center, and with the same radius, describe theindefinite arc IH, cutting AB at I. 3. With / as center, and with a radius equal to str. line FGydescribe an arc intersecting the arc IH at K, 4. Draw AK. 6. Z BAK = Z CDE, and is the Z required. II. Proof Argument1. Draw FG and IK, 2. Jw^FDG^nCiIAK,DF = AL 3. nG = AK. 4. FG = IK. Reasons 1. A str. line may be drawn from any one point toany other. § 54, 15. 2. By cons. 3. By cons. 4. By cons. BOOK I 39 5. ArgumentA FDG = A lAK, 6. ZBAK = ZCDE. Reasons 5. Two A are equal if the three sides of one areequal respectively to thethree sides of the other.§ 116. 6. Homol. parts of equal figures are equal. § 110. III. Discussion This construction is always possible, and there is
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