Problem 22: Collinear points (Iran 1998)

by henderson, Jun 8, 2016, 2:50 PM

$$\bf\color{red}Problem\ 22\ $$Let $ABC$ be a triangle and $D$ be a point on the extension of side $BC$ past $C$ such that $CD = AC.$ The circumcircle of $\triangle ACD$ intersects the circle with diameter $BC$ again at $P.$ Let $BP$ meet $AC$ at $E$ and $CP$ meet $AB$ at $F.$ Prove that the points $D, E, F$ are collinear.

$\blacktriangleright$ Link to the problem
This post has been edited 5 times. Last edited by henderson, Sep 11, 2016, 7:46 PM
collinear points

Comment

0 Comments

"Do not worry too much about your difficulties in mathematics, I can assure you that mine are still greater." - Albert Einstein

avatar

henderson
Archives
- May 2017
Problem 65: IMO Shortlist 2015, G3
+ March 2017
Problem 64: Integer functional equation
Problem 63: Concurrent lines
+ February 2017
Problem 62: Sum of fractions
+ January 2017
Problem 61: Poland MO Finals 2003, Problem 3
Problem 60:
Problem 59: ELMO Shortlist 2012, G3
Problem 58: Iran 1996 inequality
Problem 57: Prime factorization
+ December 2016
Problem 56: Brazil MO 2013, Day 1, Problem 2
Problem 55: IMO Shortlist 2005, A2
+ November 2016
Problem 54: IMO Shortlist 1997, Number Theory
+ October 2016
Problem 53: Hard geometry problem
Blanchet's Theorem
Lines are concurrent at the Lemoine point
Problem 52: Iran MO 1996, 3rd round, 3rd exam, Problem 2
+ September 2016
Problem 51: Mathematical Reflections 5 (2016), O389
Radical axis problems
Problem 50: Nice geometry problem
Useful properties in inequalities
Problem 49: IZhO 2016, Day 2, Problem 2
Liberator's blog
Problem 48: The incenter lies on circumcircle
Problem 47: APMO 2002, Problem 5
+ August 2016
Problem 46: EGMO 2016, Day 2, Problem 1
Problem 45: EGMO 2016, Day 1, Problem 2
+ July 2016
Problem 44: Nice sequence
Problem 43: Balkan MO Shortlist 2003
Problem 42: IMO Shortlist 2012 , G2
Problem 41: PEN, Diophantine Equations, Problem 13
Problem 40: PEN, Diophantine Equations, Problem 14
Parallelogram Isogonality Lemma
More Characterizations of Tangential Quadrilaterals
Problem 39: IMO 2016, Day 1, Problem 1
Problem 38: Beautiful inequality
Problem 37: Turkmenistan National MO 2016, Day 2, Problem 1
Problem 36: PEN, Linear Recurrences, Problem 5
Problem 35: Balkan MO 2016, Problem 2
Problem 34: Romania TST 2004, Problem 11
Problem 33: PEN, Recursive Sequences, Problem 6
Problem 32: PEN, Linear Recurrences, Problem 13
Problem 31: PEN, Linear Recurrences, Problem 12
+ June 2016
Problem 30: Germany 2003 (PEN A, Problem 112)
Problem 29: Useful inequality by Vasile Cirtoaje
Problem 28: Sequence
Problem 27: Inequality by Vasile Cirtoaje (2005)
Problem 26: Number of divisors
Problem 25: IMO Shortlist 2006, G3
Problem 24: Another inequality
Problem 23: Inequality
The SOS Theorem
Problem 22: Collinear points (Iran 1998)
Problem 21: Hard inequality
Problem 20: Concyclic points
Problem 19: Balkan MO Shortlist 2007
Problem 18: APMO 2016, Problem 3
Properties related to the orthocenters of BPC, CPA, APB
Problem 17: Problem related to a tangential quadrilateral
Newton-Gauss line
Problem 16: The extended of Newton line
Problem 15: The converse of the butterfly theorem works here
+ May 2016
Problem 14: The third of the parallel lines
Nice trigonometric identities
Useful form of Schur inequality
Problem 13: IMO 1987, Day 2, Problem 4
Problem 12: Vasc inequality
Problem 11: Collinear points (Bosnia-Herzegovina TST 2016)
Problem 10: Sequence & inequality
+ February 2016
Problem 9:Notes on Eucledian Geometry,radical axis,problem 1
Problem 8:Notes on Eucledian Geometry,radical axis,problem 2
Problem 7: Wonderful inequality
Problem 6: Hard inequality
Problem 5: Balkan MO 1996, Problem 3
Problem 4: Divisibility
Problem 3: Nice inequality
Problem 2: Power of 2 modulo 9
Problem 1: EGZ theorem
Shouts
Submit
7 shouts
Tags
geometry
Inequality
number theory
Sequence
inequalities
the PEN Project
IMO Shortlist
concurrent lines
functional equation
Functional Equations
incenter
radical axis
Tangents
Vasc inequality
APMO
Brokard theorem
butterfly theorem
Combinatorial Number Theory
concyclic points
Diophantine Equations
infinitely many
Linear Recurrences
Newton line
perpendicular lines
poles and polars
tangential quadrilateral
The SOS Theorem
angle bisector
Blanchets Theorem
circumcircle
collinear points
combinatorics
complex numbers in geometry
Divisibility
Divisors
ELMO Shortlist
incircle
induction
Inversion
Iran 1996 inequality
isogonality
izho
Justin Stevens
Lemma
Lemoine point
National MO
parallel
Parallel Lines
parallelogram
Perfect Squares
Poland MO
prime numbers
projective geometry
Recursive Sequences
reflection
Schur inequality
Spiral Similarity
symmedian
Trigonometric Identities
trigonometry
USAMO
About Owner
  • Posts: 312
  • Joined: Mar 10, 2015
Blog Stats
  • Blog created: Feb 11, 2016
  • Total entries: 77
  • Total visits: 20933
  • Total comments: 32
Search Blog
a