9) Let (C) be a circle of center O and diameter [AB] such that AB = 10 cm. Let C be a point on [AB] such that AC = 6 cm. Draw the circles ($C_1$) of diameter [AC] and ($C_2$) of diameter [BC]. Let D be a point on (C) such that BD = 5 cm. The line (AD) cuts ($C_1$) at E and the line (BD) cuts ($C_2$) at F. a) Show that the triangles ADB and AEC are right triangles. b) Prove that the lines (BD) and (CE) are parallel. Are the lines (CF) and (AD) parallel? c) Calculate EC and AE. Deduce that ED = $2\sqrt{3}$. d) What is the nature of the quadrilateral CEDF? Calculate its perimeter.
Added by Isaac H.
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Show that triangles ADB and AEC are right triangles: Since AB is the diameter of circle (C), angle AOB is a right angle. Since AC is the diameter of circle (C1), angle AOC is a right angle. Therefore, triangles ADB and AEC are right triangles. Show more…
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