Engineering Calculus Notes 108

Engineering Calculus Notes 108 - s is the semiperimeter s =...

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96 CHAPTER 1. COORDINATES AND VECTORS From this, prove Heron’s formula in the obtuse case: A ( ABC ) = a 2 r c 2 ( a 2 + b 2 ) 2 c ( Hint: First ±nd CD , then use the standard formula.) (b) Acute case: Suppose the angle at C is acute. Let D be the foot of the perpendicular from A to BC . Show that c 2 = a 2 + b 2 2 c · CD. From this, prove Heron’s formula in the acute case: A ( ABC ) = a 2 r ( a 2 + b 2 ) c 2 2 c 16. Heron’s Second Formula: Prove Heron’s second (and more famous) formula for the area of a triangle: A = R s ( s a )( s b )( s c ) (1.27) where a , b and c are the lengths of the sides of the triangle, and
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Unformatted text preview: s is the semiperimeter s = 1 2 ( a + b + c ) . Refer to Figure 1.51 ; we follow the exposition in [ 5 , p. 186]: The original triangle is △ ABC . (a) Inscribe a circle inside △ ABC , touching the sides at D , E , and F . Denote the center of the circle by O ; Note that OE = OF = OD. Show that AE = AF CE = CD BD = BF. ( Hint: e.g. , the triangles △ OAF and △ OAE are similar—why?)...
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