Collinearity/Concurrence∗

Ray Li (rayyli@stanford.edu)
June 29, 2017

1 Introduction/Facts you should know

1. (Cevian Triangle) Let ABC be a triangle and P be a point. Let lines AP , BP , CP
meet lines BC, AC, AB at D, E, F , respectively. Triangle DEF is called a cevian
triangle of P with respect to ABC.
2. (Ceva) Let D, E, and F be points on sides BC, AC and AB, respectively, of triangle
BD CE AF
ABC. Then AD, BE, and CF are concurrent if and only if · · = 1.
CD AE BF
3. (Trig Ceva) In triangle ABC, let D, E, and F be points on sides BC, AC, and AB
respectively. Then AD, BE and CF are concurrent if and only if
sin ∠BAD sin ∠CBE sin ∠ACF
· · = 1.
sin ∠ABE sin ∠BCF sin ∠CAD
4. (Ceva in a circle) Let A, B, C, D, E, F be six consecutive points on a circle. We have
AB CD EF
AD, BE, CF are concurrent if and only if · · = 1.
BC DE F A
5. (Menelaus) Let ABC be a triangle and let D, E, and F be points on lines BC, AC, and
BD CE AF
AB, respectively. Then D, E, and F are collinear if and only if · · = −1.
DC EA F B
6. (Isogonal conjugates) Let ABC be a triangle and P be a point not equal to any of
A,B, C. The reflections of lines AP , BP , CP over the angle bisectors of A, B, C,
respectively, concur at a point. This point is called the isogonal conjugate of ABC.
7. (Harmonic conjugates) Let ABC be a triangle and DEF be a cevian triangle. Let
P B DC
EF ∩AB = P . Then P, D are harmonic with respect to B, C. That is · = −1.
P C DB
8. (Desargues) Let A1 B1 C1 , A2 B2 C2 be triangles in space. Lines A1 A2 , B1 B2 , C1 C2
are concurrent (or all parallel) if and only if the intersections of corresponding sides
A1 A2 ∩ B1 B2 , A2 A3 ∩ B2 B3 , and A3 A1 ∩ B3 B1 are collinear.
∗
Some material and problems taken from MOP 2012 collinearity and concurrency handout by Carlos
Shine and http://www.math.cmu.edu/~ploh/docs/math/6-concur-solns.pdf

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 9. (Pappus) Let A1 , A2 , A3 and B1 , B2 , B3 be two sets of collinear points. Then A1 B2 ∩
A2 B1 , A1 B3 ∩ A3 B1 , A2 B3 ∩ A3 B2 are collinear.

10. (Pascal) Let ABCDEF be six points on a conic. Then the intersections of AB ∩ DE,
BC ∩ EF , and CD ∩ F A are collinear. The hexagon does not need to be convex, and
degenerate cases are allowed. For example, if we took the hexagon AABCDE, then
AA is the tangent through A.

11. (Radical center) Let ω1 , ω2 , ω3 be circles. Then the radical axes of ω1 , ω2 , ω2 , ω3 , and
ω3 , ω1 are either all parallel or concurrent at the radical center of the three circles.

12. (Brianchon) Let ABCDEF be a hexagon circumscribed to a circle. Then AD, BE, CF
are concurrent. The hexagon does not need to be convex, and degenerate cases allowed.

2 Warmups
1. Prove the existence of Isogonal conjugates.

2. (Gergonne Point) Let ABC be a triangle. Let A1 , B1 , C1 be the points where the incircle
touches sides BC, AC and AB, respectively. Prove AA1 , BB1 , CC1 are concurrent.

3. Let ABC be a triangle with AB = AC. Let ` be a line that intersects sides BC, AC,
and AB at D, E, and F respectively. Suppose that DE = DF. Show that CE = BF.

4. Let ABC be a triangle. A line through M , the midpoint of BC, parallel to the angle
bisector of ∠A meets sides AC and AB at E and F, respectively. Show that CE = BF.

5. ABCD trapezoid with AB||CD. AC and BD meet at P and AD and BC meet at Q.

Show that P Q passes through the midpoints of AB and CD.

6. (Euler line) Prove the circumcenter, orthocenter, and centroid of a triangle are collinear.

3 Easier Problems
7. (Cevian Nest Theorem) Let 4Q1 Q2 Q3 be the cevian triangle of a point P with respect
to 4P1 P2 P3 , and let 4R1 R2 R3 be the cevian triangle of a point Q with respect to
4Q1 Q2 Q3 . Prove that lines P1 R1 , P2 R2 , P3 R3 are concurrent.

8. (IMO 1961) Let D, E, and F be points on sides BC, AC and AB, respectively, of
triangle ABC, such that AD, BE, and CF are concurrent at point P. Show that
among the numbers PAPD , BP , CP , at least one is ≥ 2 and at least one is ≤ 2.
PE PF

9. (Own) In cyclic quadrilateral ABCD, AB ∗ BC = AD ∗ DC. Let X, Y be on CD, BC

respectively such that BX||AD, DY ||AB. Prove XY ||BD.

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10. (USAMO 2003) Let ABC be a triangle. A circle passing through A and B intersects
segments AC and BC at D and E, respectively. Lines AB and DE intersect at
F , while lines BD and CF intersect at M . Prove that M F = M C if and only if
M B · M D = M C 2.

11. (IMO Shortlist 2001) Let A1 be the center of the square inscribed in acute triangle ABC
with two vertices of the square on side BC. Thus one of the two remaining vertices
of the square is on side AB and the other is on AC. Points B1 , C1 are defined in a
similar way for inscribed squares with two vertices on sides AC and AB, respectively.
Prove that lines AA1 , BB1 , CC1 are concurrent.

12. Let ABC be a triangle and A1 , B1 , C1 be the points where the incircle touches sides
BC, AC, and AB, respectively. Prove that if A2 , B2 , C2 are points on minor arcs
B1 C1 , C1 A1 , A1 B1 , respectively, of the incircle of ABC such that AA2 , BB2 , CC2 are
concurrent, then A1 A2 , B1 B2 , C1 C2 are concurrent.

13. (Russia 1997) Given triangle ABC, let A1 , B1 , C1 be the midpoints of the broken lines
CAB, ABC, BCA, respectively. Let `A , `B , `C be the respective lines through A1 , B1 ,
C1 parallel to the angle bisectors of A, B, C. Show that `A , `B , `C are concurrent.

14. (IMO 1982) Let ABCDEF be a regular hexagon, with points M and N on diagonals
AC and CE respectively, such that AM
AC
= CN
CE
= r. Find r, if B, M, N are collinear.

15. (Seven Circles Theorem) Let C, C1 , C2 , C3 , C4 , C5 , C6 be circles such that C is externally

tangent to Ci at Pi for all i, and Ci is externally tangent to Ci+1 for all i, with C6
externally tangent to C1 . Prove that P1 P4 , P2 P5 , P3 P6 are concurrent.

16. (Classic) Let ABC be a triangle with incenter I. Let Γ be the circle tangent to sides
AB, AC, and the circumcircle of ABC. Let Γ touch sides AB and AC at X and Y ,
respectively. Prove I is the midpoint of XY .

17. (Composition of homotheties) If σ1 , σ2 are homotheties with centers O1 , O2 respectively

and ratios k1 , k2 (possibly negative), and k1 k2 6= 1, then the composition σ1 ◦ σ2 is a
homothety with center O and ratio k1 k2 and O, O1 , O2 are collinear. Try proving this
with Desargues’ theorem.

18. (Monge’s Theorem) Let ω1 , ω2 , ω3 be three circles such that no circle contains another
circle. Let P1 be the intersection points of the common external tangents of ω2 and
ω3 , and define P2 , P3 similarly. Show that P1 , P2 , and P3 are collinear.

4 Problems
19. (MOP 1998) Let ABC be a triangle, and let A0 , B 0 , C 0 be the midpoints of the arcs
BC, CA, AB, respectively, of the circumcircle of ABC. The line A0 B 0 meets BC and
AC at S and T . B 0 C 0 meets AC and AB at F and P , and C 0 A0 meets AB and BC at
Q and R. Prove that the segments P S, QT , F R concur.

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20. (Bulgaria 1997) Let ABCD be a convex quadrilateral such that ∠DAB = ∠ABC =
∠BCD. Let H and O denote the orthocenter and circumcenter of the triangle ABC.
Prove that H, O, D are collinear.

21. (IMO Shortlist 1997) The bisectors of angles A, B, C of triangle ABC meet its cir-
cumcircle again at the points K, L, M , respectively. Let R be an internal point on
side AB. The points P and Q are defined by the conditions: RP is parallel to AK and
BP is perpendicular to BL; RQ is parallel to BL and AQ is perpendicular to AK.
Show that the lines KP , LQ, M R concur.

22. Let ABC be a triangle, D, E, F be the points of tangency of the incircle with BC,
CA, and AB, respectively, and A0 , B 0 , C 0 be the midpoints of arcs BC, AC, and AB,
respectively, on the circumcircle. If I is the incenter, and M, N, P are the midpoints
of the segments ID, IE, IF , show that lines M A0 , N B 0 , P C 0 pass through the same
point.

23. (IMO Shortlist 1997) Let A1 A2 A3 be a non-isosceles triangle with incenter I. Let Ci ,
i = 1, 2, 3, be the smaller circle through I tangent to Ai Ai+1 and Ai Ai+2 (indices mod
3). Let Bi , i = 1, 2, 3, be the second point of intersection of Ci+1 and Ci+2 . Prove that
the circumcenters of the triangles A1 B1 I, A2 B2 I, A3 B3 I are collinear.

24. (IMO Shortlist 2006) Circles ω1 and ω2 with centres O1 and O2 are externally tangent
at point D and internally tangent to a circle ω at points E and F respectively. Line t
is the common tangent of ω1 and ω2 at D. Let AB be the diameter of ω perpendicular
to t, so that A, E, O1 are on the same side of t. Prove that lines AO1 , BO2 , EF and
t are concurrent.

25. (TSTST 2017) Let ABC be a triangle with incenter I. Let D be a point on side BC
and let ωB and ωC be the incircles of 4ABD and 4ACD, respectively. Suppose that
ωB and ωC are tangent to segment BC at points E and F , respectively. Let P be the
intersection of segment AD with the line joining the centers of ωB and ωC . Let X be
the intersection point of lines BI and CP and let Y be the intersection point of lines
CI and BP . Prove that lines EX and F Y meet on the incircle of 4ABC.

26. Three equal circles ω1 , ω2 , ω3 are tangent to the pairs of sides of triangle ABC that
meet at A, B, and C, respectively, and are also internally tangent to the circle Ω.
Prove that the center of Ω lies on the line that passes through the incenter and the
circumcenter of ABC.

27. Let ω be a circle and ABCD be a square somewhere in its interior. Construct the
circles ωa , ωb , ωc , ωd to be exterior to the square but internally tangent to the circle ω,
at some points denoted by A0 , B 0 , C 0 , D0 , respectively, and such that they are tangent
to the pairs of lines AB and AD, AB and BC, BC and DC, and CD and DA,
respectively. Prove that the lines AA0 , BB 0 , CC 0 , DD0 meet at one point.

28. (IMO Shortlist 2007) Point P lies on side AB of a convex quadrilateral ABCD. Let ω
be the incircle of triangle CP D, and let I be its incenter. Suppose that ω is tangent

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 to the incircles of triangles AP D and BP C at points K and L, respectively. Let lines
AC and BD meet at E, and let lines AK and BL meet at F . Prove that points E, I,
and F are collinear.

29. (IMO 2008) Let ABCD be a convex quadrilateral with AB 6= BC. Denote by ω1
and ω2 the incircles of triangles ABC and ADC. Suppose that there exists a circle
ω inscribed in angle ABC, tangent to the extensions of line segments AD and CD.
Prove that the common external tangents of ω1 and ω2 intersect on ω.