V. Cyclic Quadrilaterals
June 24, 2003
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
June 20, 2003
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,
III. Telescoping Sums and Products
June 18, 2003
Trig (stolen from Titu97)
(2k + 1)2
Solution: If an is a positive sequence, then:
an = tan
1 + an+1 an
This telescopes with an = 2n from th
VI. Collinearity and Concurrence
June 26, 2003
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
F B DC EA
Trig Ceva Let ABC be a triangle, and let D BC, E CA, and
Red MOP Lecture: June 20, 2002.
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
June 16, 2003
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.
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
1. Extremely ruthless or cruel.
2. Crude or unfeeling in manner or
3. Harsh; unrelenting.
4. Disagreeably precise or penetrating.
1. The capacity to do work or cause physic
June 18, 2003
(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
Heres the e
Red MOP Lecture (with solutions): July 16, 2013
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