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Engineering Calculus Notes 109

Engineering Calculus Notes 109 - (e It then follows by...

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1.6. CROSS PRODUCTS 97 O D C E A F B L H J Figure 1.51: Heron’s Formula (b) Show that the area of ABC equals s · OD . ( Hint: Consider OBC , OAC and OAB .) (c) Extend CB to H , so that BH = AF . Show that s = CH. (d) Let L be the intersection of the line through O perpendicular to OC with the line through B perpendicular to BC . Show that the points O , B , L and C all lie on a common circle. ( Hint: Each of the triangles CBL and COL have right angles opposite their common edge CL , and the hypotenuse of a right triangle is a diameter of a circle containing the right angle.)
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Unformatted text preview: (e) It then follows by Proposition III.22 of the Elements (opposite angles of a quadrilateral inscribed in a circle sum to two right angles) that ∠ CLB + ∠ COB equals two right angles. Show that ∠ BOC + ∠ AOF equals two right angles. ( Hint: Each of the lines from O to a vertex of △ ABC bisects the angle there.) It follows that ∠ CLB = ∠ AOF....
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