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Geom8 feb 6

# Geom8 feb 6 - PEM Level C 6 February 2010 Topics in...

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PEM Level C 6 February 2010 Topics in Geometry: Menelaus’ Theorem Theorems: 1. (Menelaus’ Theorem) Let ABC be a triangle, and let X , Y , and Z be the feet of some cevians from A , B , and C , respectively. If X , Y , and Z are collinear, then AZ ZB · BX XC · CY Y A = 1 . 2. (Converse of Menelaus’ Theorem) Let ABC be a triangle, and let X , Y , and Z be the feet of some cevians from A , B , and C , respectively. If AZ ZB · BX XC · CY Y A = 1 , then X , Y , and Z are collinear. Exercises: 1. Triangle ABC has B = 90 , BC = 3, and AB = 4. Point D is on segment AC such that AD = 1, and point E is the midpoint of AB . Join D and E by a segment, and extend DE to meet the extended CB at F . Find BF . 2. In 4 ABC , let X and Y be points on segments BC and AC , respectively, and R the point of intersection of AX and BY . Given that AY/Y C = p and AR/RX = q , where 0 < p < q , express BX : XC in terms of p and q . 3. The diagonals AC and BD of the convex quadrilateral ABCD intersect at point M in such a way that AM = MC and DM = 2 MB . Suppose that X and Y
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