V. Cyclic Quadrilaterals
Po-Shen Loh
June 24, 2003
1
All You Need To Know (sort of )
A quadrilateral is cyclic if and only if the sum of a pair of opposite angles is 180.
A quadrilateral is cyclic if and only if it satisfies power of a point; that is, i
IV. Triangles
Po-Shen Loh
June 20, 2003
1
Warm-up
1. (Greece) Let ABC be a triangle, O be the foot of th eangle bisector of A, and K the second intersection
of AO with the circumcircle of ABC. Prove that if the incircles of BOK and COK are congruent,
then
III. Telescoping Sums and Products
Po-Shen Loh
June 18, 2003
1
Trig (stolen from Titu97)
1. Evaluate:
X
tan1
k=0
2
(2k + 1)2
Solution: If an is a positive sequence, then:
1
tan
1
an+1 tan
1
an = tan
an+1 an
1 + an+1 an
This telescopes with an = 2n from th
VI. Collinearity and Concurrence
Po-Shen Loh
June 26, 2003
1
Your Weapons
Ceva Let ABC be a triangle, and let D BC, E CA, and F AB. Then AD, BE, and CF concur if
and only if:
AF BD CE
= 1.
F B DC EA
Trig Ceva Let ABC be a triangle, and let D BC, E CA, and
Red MOP Lecture: June 20, 2002.
Po-Shen Loh
1
Common abbreviations for geometry problems
Given triangle ABC:
a, b, and c are the lengths of the sides opposing vertices A, B, and C, respectively.
s is the semiperimeter
r is the inradius
R is the circum
I. Induction
Po-Shen Loh
June 16, 2003
1
News Flash From Zuming!
Remind Po to take all the markers from CBA 337
Tonights study session for Red/Blue is in Bessey 104
Future Red lectures are in NM B-7, the Naval Military Base #7
Future Red tests and stu
Yellow MOP Lecture: 910 July 2002.
Po-Shen Loh
1
Triangles
1.1
Facts
1. Extended Law of Sines a/ sin A = 2R.
2. [ABC] = abc/4R.
3. (Geometry Revisited, page 3.) Let p and q be the radii of two circles through A, touching BC at B
and C, respectively. Then
Brutal Force II
Po-Shen Loh MOP 2002 11 July 2002
brutal (adj.)
1. Extremely ruthless or cruel.
2. Crude or unfeeling in manner or
speech.
3. Harsh; unrelenting.
4. Disagreeably precise or penetrating.
force (n.)
1. The capacity to do work or cause physic
II. Inequalities
Po-Shen Loh
June 18, 2003
1
Warm-Up
(Po98) Prove that for all ordered triples (a, b, c) of prime numbers:
a2 b + a2 + ac2 + 115a + b2 c + b2 + c2 + 27c + 176 < 6ab + 22ac + 14bc + 5b.
Solution: Complete the square
2
The p-Norm
Heres the e
Red MOP Lecture (with solutions): July 16, 2013
Po-Shen Loh
1
Definitions
Definition 1 Let be a circle with center O and radius r, and let P be a point. Then the power of P with
respect to is OP 2 r2 . Note that the power can be negative.
Definition 2 Let