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 t
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
a
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
T
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 i
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-
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
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
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.
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
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 i
Beginde Lubuk Gong
Cerita Rakyat Sumatra Selatan
Pada zaman dahulu kala, di daerah Sumatra Utara, ada
seorang beginde (kepala desa) yang kaya raya bernama
Beginde Lubuk Gong. Ia sangat ketat menjalank