PEM Level C
21 November 2009
Topics in Geometry - Basic Concepts and Theorems
Examples: 1. (PMO 1984-85) Let the rectangle ABCD be such that AB = CD = 4 units and BC = AD = 3 units. A perpendicular is dropped to the diagonal BD from each of the vertices A
Practice problems for sections on September 27th and 29th.
Here are some example problems about the product, fraction and chain rules for derivatives and implicit differentiation. If you notice any errors please let me know.
1. (easy) Find the equation of
PEM Level C
19 February 2010
Trigonometric Identities
Exercises: 1. Let ABC be an acute-angled triangle. Prove that tan A+tan B +tan C = tan A tan B tan C. 2. Let ABC be a triangle. Prove that sin2 A B C 3 + sin2 + sin2 . 2 2 2 4
3. Let ABC be a triangle
PEM Level C
12 February 2010
Trigonometry: Law Cosines and Sines
Theorems: 1. (Law of Cosines) In ABC , the following equations hold: a2 = b2 + c2 2bc cos A b2 = a2 + c2 2ac cos B c2 = a2 + b2 2ab cos C 2. (Extended Law of Sines) Let R be the circumradius
PEM Level C
6 March 2010
Problem Solving
1. (IMO 1968) Prove that there is one and only one triangle whose lengths of its sides are consecutive integers, and one of whose angles is twice as large as another. 2. (IMO 1976) In a convex quadrilateral of area
PEM Level C
26 February 2010
Geometric Inequality
(Triangle Inequality) Let A, B , and C be points on the plane. Then AB + BC AC ; where equality holds if and only if B lies on the segment AC . (Side-Angle Inequality) In Exercises: 1. Let a, b, and c be
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 BX CY = 1.
PEM Level C
16 January 2010
Topics in Geometry: More on Triangles
Theorems: 1. The perpendicular bisectors of the sides of a triangle are concurrent. 2. (Brahmaguptas Theorem) The product of the lengths of two sides of a triangle is equal to the product o
PEM Level C
9 January 2010
Topics in Geometry - Power of a Point and the Radical Axis
Theorems: 1. Let P be any point, and be a given circle. Let points A and B . be a line through P that intersects at
(a) If the P is inside , then P(P, ) = P A P B . (b)
PEM Level C
12 December 2009
Topics in Geometry - The Power Theorems, Power of a Point
Theorems: 1. (Power Theorem) Let P be a given point, and let be a line that passes through P and intersecting a given circle at two (not necessarily distinct) points A
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 se
PEM Level C
28 November 2009
Topics in Geometry - Chords, Tangents and Intersecting Circles
Examples: 1. In a circle, chord AB is bisected at E by chord CD. If CE = 16 and ED = 4, nd the length of chord AB . 2. An arch is built in the form of an arc of a
PEM Level C
9 January 2010
Topics in Geometry - All about circles
Exercises: 1. Prove that a line cannot cut a circle at more than two points. 2. Two circles intersect at two points. Prove that the length of the line segment passing through one point of i
Review Problems
1. Find the solution set of the following equations or inequalities.
(a) x2 + 3x 11 < x 3
13 x
x1
(b) |x| |2x 3| = x 3
(c) |2x2 3x 35| |x 5|
2. Given the points A(3, 1), B(2, 2) and C(5, 1), determine the coordinates of point D such
that