Geom3 dec 5

# Geom3 dec 5 - the other two vertices are equal . Exercises:...

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PEM Level C 5 December 2009 Topics in Geometry - Cyclic Quadrilaterals Examples: 1. Theorem: A convex quadrilateral is cyclic if and only if two opposite angles are supplementary . 2. Triangle ABC is isosceles with AB = AC . Let D and E be points on the segments AB and AC , respectively, such that DE k BC . Prove that EDBC is cyclic. 3. Let ABCD be a cyclic quadrilateral such that the rays AB and DC intersect at the point X . Show that 4 XAD and 4 XCB are similar. 4. Theorem: (Ptolemy’s Theorem) A convex quadrilateral ABCD is cyclic if and only if AB · CD + AD · BC = AC · BD . 5. Let X be an arbitrary point on the minor arc BC of the circumcircle of an equilateral triangle ABC . Prove that AX = BX + CX . 6. Theorem: A convex quadrilateral is cyclic if and only if the angles subtended by any side at

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Unformatted text preview: the other two vertices are equal . Exercises: 1. In convex quadrilateral ABCD , the diagonals AC and BD meet at point E . Given that AE = 2, BE = 5, CE = 10, DE = 4 and BC = 7 . 5, ﬁnd AB . 2. Prove that, in a cyclic quadrilateral, two opposite sides are equal if and only if the other two opposite sides are parallel. 3. Determine the values of the numbered angles. 4. Name the cyclic quadrilaterals in the following diagrams. Explain your reasoning. 5. Determine the measures of angles 1 through 5 (below, left). 6. If ABCD is an isosceles trapezoid, prove that ABCD is cyclic....
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## Geom3 dec 5 - the other two vertices are equal . Exercises:...

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