Projective Geometry

7/21/2019 Projective Geometry

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Projective Geometry

Michael Kural

July 4, 2015

1 Reference Sheet

1.1 Projective Plane

We first define the projective plane. It’s nearly the same as the normal euclideanplane, but it’s much nicer to work with because it accommodates stupid special cases.From the perspective of concision and laziness, you should pretty much always usethe projective plane.

Definition 1.1. We define the projective plane RP 2 as follows: to the ordinaryEuclidean plane R2, append a line at infinity , which we imagine to encircle theplane in a type of infinite loop. The line consists of points at infinity , each of whichcorresponds to a direction or slope (possibly infinite!) in the plane. Each point lieson all lines with this slope, and the points at infinity all lie on the line at infinity.

Note then that our original visualization was slightly misleading: the line at

infinity isn’t a full loop, because we imagine a Euclidean line to stretch infinitely farin both directions, but then meet again at a single point: so the line at infinity ismore like half of this loop.

We can also define the real projective plane algebraically as

R3 − (0, 0, 0)

modded out by the equivalence relation

v ∼ kv, k ∈ R.

This is more symmetric, and gives rise to the former definition if we do casework on

whether or not the last entry is 0. We can also define it as the sphere modded outby the equivalence relation associating a point with its antipode. However, this isa talk on olympiad geometry, so we will focus on the first definition, which is morehelpful in this context.

Now, the good thing is that the projective plane gives us a context in which todevelop the rest of olympiad projective geometry without having to worry about

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Michael Kural Projective Geometry

special cases. For example, theorems sometimes conclude that three lines are eitherconcurrent or parallel, while a proof might handle an isosceles case separately fromthe general case. In the first example, we interpret parallelism to be the same as

concurrency (because the lines concur at a point at infinity). In the second, we cansimply deal with points defined at infinity in the same way we would normal points.Another plus is that we can rigorously use points at infinity in nice ways (as will beshown) without having to make some kind of limiting argument each time.

1.2 Cross Ratio and Harmonic Divisions

Definition 1.2. The cross ratio of four collinear points A , B, C , D, denoted (A, B; C, D),is the directed ratio

(A, B; C, D) = AC · BD

BC · AD

Definition 1.3. Four collinear points A, B, C, D constitute a harmonic division (alternatively, they are harmonic ) if their cross ratio equals −1.

(A, B; C, D) = −1 ⇐⇒ AC

BC = −

AD

BD

Lemma 1. Suppose A, B, C, D are four points on a line 1, and P is a point not on this line, while 2 is another line in the plane. Suppose P A, P B , P C, P D intersect 1 at A, B, C , D. Then (A, B; C, D) = (A, B; C , D). In particular, if the first quadruple is harmonic, then so is the second.

Note, then, that four lines through a given point can be characterized as har-monic independently of the new line they intersect.

Definition 1.4. If four lines P A, P B , P C, P D pass through P and form a harmonicdivision with each line they intersect, the four lines are called a harmonic pencil ,denoted P (A, B; C, D) or (P A , P B; P C , P D).

Lemma 2. In triangle A, B, C , suppose that AD,BE,CF are concurrent cevians (with D , E , F on lines BC,CA,AB, respectively), and D, E , F are collinear with D on BC . Then (D, D; B, C ) is harmonic.

Example 1.1. If ABC is a triangle, and D, D are the feet of the external andinternal bisectors, then (D, D; B, C ) is harmonic.

Example 1.2. If ABC is a triangle, and M is the midpoint of BC , while is theline through A parallel to BC , then (AB,AC ; AM,) is harmonic. In other words,

if P ∞ is the point at infinity in the direction of line BC , then (B, C ; M, P ∞) isharmonic.

Here is an alternative characterization.

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1.4 Miscellaneous Theorems

Theorem 2 (Pascal’s Theorem). Six points A, B, C, D, E , F in a plane lie on a

conic if and only if AB ∩ DE,BC ∩ EF, and CD ∩ F A are collinear.

Remark. The most common case for which Pascal is used is when the conic is acircle (so the 6 points are concyclic). When the conic degerates to a pair of lines,we recover Pappus’ Theorem.

Remark. Although the points should be distinct for the theorem to make sense, inpractice we can apply Pascal’s theorem on a degenerate hexagon by considering theline AA to be the tangent line from A to the common conic. (This is more rigorousif you consider the limit of Pascal’s theorem as one point approaches a second.)

Theorem 3 (Pappus). Suppose points A1, A2, A3 are collinear and B1, B2, B3 arecollinear. Then A1B2 ∩ A2B1, A1B3 ∩ A3B1, and A2B3 ∩ A3B2 are collinear.

Theorem 4 (Brianchon). Suppose ABCDEF is a circumscribed hexagon. ThenAD,BE, and CF concur.

Definition 1.9. Two triangles A1A2A3 and B1B2B3 are perspective from a point if A1B1, A2B2, and A3B3 concur.

Definition 1.10. Two triangles A1A2A3 and B1B2B3 are perspective from a line if A1A2 ∩ B1B2, A2A3 ∩ B2B3, and A3A1 ∩ B3B1 are collinear.

Theorem 5 (Desargues’ Theorem). Two triangles are perspective from a point if and only if they are perspective from a line. In this case, we say the two triangles

are simply perspective .Definition 1.11. The exsimilicenter of two circles Γ1 and Γ2 is the point P suchthat there exists a positive homothety centered at P mapping Γ1 to Γ2. The insim-ilicenter is definited similarly, but for a negative homothety. Generally, the exsim-ilicenter is the intersection of the two external tangents, while the insimilicenter of the intersection of the two internal tangents (if such tangents exist).

Theorem 6 (Monge’s Theorem). Given three circles in the plane, their three pair-wise exsimilicenters are collinear. Additionally, an exsimilicenter and two insimili-centers are collinear.

Theorem 7 (Brokard’s Theorem). Suppose ABCD is a cyclic quadrilateral. LetP = AB ∩CD,Q = BC ∩AD, and AC ∩BD. Then P QR is self-polar; that is, QR

is the polar of P , P R is the polar of Q, and P Q is the polar of R.

Definition 1.12. An involution (also known as an involutive homography ) on a line is transformation on that line, which is an inversion about some point P on withpositive or negative radius (see the definition of inversion above).

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Theorem 8 (Desargues Involution Theorem). Let A, B, C , D be four fixed points,and a fixed line. If varying conic through A, B, C, D intersects at two pointsP, Q, then a fixed involution maps each P to Q. In particular, a pair of linesis a conic, so if AB, CD,AC, BD, AD,BC intersect at X, X , Y , Y , Z , Z , thenX, X , Y, Y , Z, Z are corresponding pairs in a common involution.

2 Problems

These are arranged roughly in order of difficulty.

1. Prove that four points in the complex plane are concyclic if and only if theircross ratio is real (cross ratio is defined using differences of complex numbersin this case).

2. Let ABC be a triangle, with A-altitude AD, and let X be an arbitrary pointon AD. Let BX meet AC at P and C X meet AB at Q. Show that AD bisects∠P DQ.

3. (USAJMO 2011) Points A, B, C , D, E lie on a circle ω and point P lies outsidethe circle. The given points are such that (i) lines P B and P D are tangent toω, (ii) P, A, C are collinear, and (iii) DE AC . Prove that BE bisects AC .

4. (Lemoine Line) Let ABC be a triangle, and let D be the point on BC such thatAD is tangent to its circumcircle. Define E, F similarly. Show that D , E , F

are collinear.

5. (IMO 2003) Let ABCD be a cyclic quadrilateral. Let P , Q, R be the feet of

the perpendiculars from D to the lines B C , C A, AB, respectively. Show thatP Q = QR if and only if the bisectors of ∠ABC and ∠ADC are concurrentwith AC .

6. You are given a line segment in the plane. The line which contains the segmentis cut off by a lake. You cannot draw lines through the lake, but you want todraw the portion of the line containing the segment which emerges from theother side of the lake. How can you construct this using only a ruler?

7. (PUMaC 2012) Let ABC be a triangle with incenter I , and let D be the footof the angle bisector from A to BC . Let Γ be the cirumcircle of triangle BIC ,and let P Q be a chord of Γ passing through D. Prove that AD bisects ∠P AQ.

8. (PUMaC 2013) Let γ be the incircle of ABC (i.e. the circle inscribed inABC ) and I be the center of γ . Let D, E and F be the feet of the per-pendiculars from I to BC , CA, and AB respectively. Let D be the pointon γ such that DD is a diameter of γ . Suppose the tangent to γ through D

intersects the line EF at P . Suppose the tangent to γ through D intersectsthe line EF at Q. Prove that ∠P IQ + ∠DAD = 180.

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9. Let ABCD be a cyclic quadrilateral, and let M, N be the midpoints of AC,BD.Suppose BD is the bisector of ∠ANC ; prove that AC is also the bisector of ∠BM D.

10. (IMO Shortlist 1998) Let I be the incenter of triangle ABC . Let K, L andM be the points of tangency of the incircle of ABC with AB,BC and CA,respectively. The line t passes through B and is parallel to KL. The linesM K and M L intersect t at the points R and S . Prove that ∠RIS is acute.

11. (IMO Shortlist 1998) Let ABC be a triangle such that ∠A = 90 and ∠B <

∠C . The tangent at A to the circumcircle ω of triangle ABC meets the lineBC at D. Let E be the reflection of A in the line BC , let X be the foot of the perpendicular from A to BE , and let Y be the midpoint of the segmentAX . Let the line BY intersect the circle ω again at Z .

Prove that the line BD is tangent to the circumcircle of triangle ADZ .

12. (IMO Shortlist 2004) Let Γ be a circle and let d be a line such that Γ and d

have no common points. Further, let AB be a diameter of the circle Γ; assumethat this diameter AB is perpendicular to the line d, and the point B is nearerto the line d than the point A. Let C be an arbitrary point on the circle Γ,different from the points A and B. Let D be the point of intersection of thelines AC and d. One of the two tangents from the point D to the circle Γtouches this circle Γ at a point E ; hereby, we assume that the points B and E

lie in the same halfplane with respect to the line AC . Denote by F the pointof intersection of the lines B E and d. Let the line AF intersect the circle Γ ata point G, different from A.

Prove that the reflection of the point G in the line AB lies on the line C F .

13. (IMO Shortlist 2014) Let Ω and O be the circumcircle and the circumcentre of an acute-angled triangle ABC with AB > BC . The angle bisector of ∠ABC

intersects Ω at M = B. Let Γ be the circle with diameter BM . The anglebisectors of ∠AOB and ∠BOC intersect Γ at points P and Q, respectively.The point R is chosen on the line P Q so that BR = M R. Prove that BR AC .(Here we always assume that an angle bisector is a ray.)

14. (Vietnam 2009) Let A, B be two fixed points and C is a variable point onthe plane such that ∠ACB = α (constant) (0 ≤ α ≤ 180). Let D, E , F

be the projections of the incenter I of triangle ABC to its sides AB, CA,BC , respectively. Denoted by M , N the intersections of AI , BI with EF ,respectively. Prove that the length of the segment M N is constant and thecircumcircle of triangle DM N always passes through a fixed point.

15. (IMO Shortlist 2004) Given a cyclic quadrilateral ABCD, let M be the mid-point of the side CD , and let N be a point on the circumcircle of triangle

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ABM . Assume that the point N is different from the point M and satisfiesAN BN

= AM BM

. Prove that the points E , F , N are collinear, where E = AC ∩BD

and F = BC ∩ DA.

16. (IMO Shortlist 2000) Let O be the circumcenter and H the orthocenter of anacute triangle ABC . Show that there exist points D, E , and F on sides BC ,CA, and AB respectively such that

OD + DH = OE + EH = OF + F H

and the lines AD, BE , and CF are concurrent.

17. (IMO 2012) Let ABC be a triangle with ∠BC A = 90, and let D be the footof the altitude from C . Let X be a point in the interior of the segment CD.Let K be the point on the segment AX such that B K = BC . Similarly, let L

be the point on the segment BX such that AL = AC . Let M be the point of intersection of AL and BK .

18. (IMO Shortlist 2002) The incircle Ω of the acute-angled triangle ABC is tan-gent to its side BC at a point K . Let AD be an altitude of triangle ABC , andlet M be the midpoint of the segment AD. If N is the common point of thecircle Ω and the line KM (distinct from K ), then prove that the incircle Ωand the circumcircle of triangle BC N are tangent to each other at the pointN .

19. (IMO Shortlist 2009) Given a cyclic quadrilateral ABCD, let the diagonalsAC and BD meet at E and the lines AD and BC meet at F . The midpoints

of AB and CD are G and H , respectively. Show that EF is tangent at E tothe circle through the points E , G and H .

20. (Romania TST 2007) Let ABC be a triangle, let E, F be the tangency points of the incircle Γ(I ) to the sides AC , respectively AB , and let M be the midpointof the side BC . Let N = AM ∩ EF , let γ (M ) be the circle of diameter BC ,and let X, Y be the other (than B, C ) intersection points of BI , respectivelyCI , with γ . Prove that

N X

N Y =

AC

AB.

21. (IMO Shortlist 2010) Three circular arcs γ 1, γ 2, and γ 3 connect the points A

and C. These arcs lie in the same half-plane defined by line AC in such away that arc γ 2 lies between the arcs γ 1 and γ 3. Point B lies on the segmentAC. Let h1, h2, and h3 be three rays starting at B, lying in the same half-plane, h2 being between h1 and h3. For i, j = 1, 2, 3, denote by V ij the point

of intersection of hi and γ j . Denote by V ijV kj V klV il the curved quadrilateral,whose sides are the segments V ijV il, V kjV kl and arcs V ijV kj and V ilV kl. We

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say that this quadrilateral is circumscribed if there exists a circle touchingthese two segments and two arcs. Prove that if the curved quadrilaterals V 11V 21 V 22V 12, V 12V 22 V 23V 13, V 21V 31 V 32V 22 are circumscribed, then the curvedquadrilateral V 22V 32 V 33V 23 is circumscribed, too.

22. (RMM 2011) A triangle ABC is inscribed in a circle ω. A variable line

chosen parallel to BC meets segments AB, AC at points D, E respectively,and meets ω at points K , L (where D lies between K and E ). Circle γ 1 istangent to the segments KD and BD and also tangent to ω, while circle γ 2is tangent to the segments LE and C E and also tangent to ω . Determine thelocus, as varies, of the meeting point of the common inner tangents to γ 1and γ 2.

23. (RMM 2012) Let ABC be a triangle and let I and O denote its incentre andcircumcentre respectively. Let ωA be the circle through B and C which istangent to the incircle of the triangle ABC ; the circles ωB and ωC are definedsimilarly. The circles ωB and ωC meet at a point A distinct from A; the pointsB and C are defined similarly. Prove that the lines AA, BB and CC areconcurrent at a point on the line IO .

24. (a) (Gergonne’s Solution to the Apollonius Problem) If three circles ω1, ω2,

and ω3 are given in the plane in sufficiently general position, prove thatthe following construction yields circle ω (actually one of 8 such circles)which is tangent to all three of the given circles:

Let be the Monge D’alembert line of the three circles (the common lineof the three exsimilicenters, or one exsimilicenter and two insimilicenters

depending on which ω we are trying to construct). Let P 1, P 2, P 3 bethe poles of with respect to ω1, ω2, ω3. Let R be the radical centerof ω1, ω2, ω3. Then RP 1, RP 2, RP 3 intersect ω1, ω2, ω3 at pairs A1, B1,A2, B2, A3, B3 such that the circumcircles of A1A2A3 and B1B2B3

are both choices for the construction of ω.

(b) (RMM 2013) Let ABCD be a quadrilateral inscribed in a circle ω. Thelines AB and CD meet at P , the lines AD and BC meet at Q, and thediagonals AC and BD meet at R. Let M be the midpoint of the segmentP Q, and let K be the common point of the segment M R and the circleω. Prove that the circumcircle of the triangle KP Q and ω are tangentto one another.

25. (Iran TST 2009) In triangle ABC , D, E and F are the points of tangency of incircle with the center of I to BC , CA and AB respectively. Let M be thefoot of the perpendicular from D to EF . P is on DM such that DP = M P .If H is the orthocenter of BI C , prove that P H bisects EF .

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26. (USA TST 2015, simplified) Suppose ABC is a triangle, and D , E , F lie onits circumcircle with AD BC, BE CA,CF AB. Let P be a point onthe Euler line of triangle ABC , and let the cevians AP,BP,CP meet thecircumcircle at X, Y , Z . Prove that DX,EY,CZ concur.

27. (USA TSTST 2014) Consider a convex pentagon circumscribed about a circle.We name the lines that connect vertices of the pentagon with the oppositepoints of tangency with the circle gergonnians.

(a) Prove that if four gergonnians are conncurrent, the all five of them areconcurrent.

(b) Prove that if there is a triple of gergonnians that are concurrent, thenthere is another triple of gergonnians that are concurrent.

28. (Iran TST 2010) Circles W 1, W 2 intersect at P, K . XY is common tangentof two circles which is nearer to P and X is on W 1 and Y is on W 2. XP

intersects W 2 for the second time in C and Y P intersects W 1 in B. Let A beintersection point of BX and CY . Prove that if Q is the second intersectionpoint of circumcircles of ABC and AXY .

29. (IMO Shortlist 2012) Let ABC be a triangle with circumcircle ω and a linewithout common points with ω. Denote by P the foot of the perpendicularfrom the center of ω to . The side-lines BC,CA,AB intersect at the pointsX , Y , Z different from P . Prove that the circumcircles of the triangles AXP ,BY P and CZ P have a common point different from P or are mutually tangentat P .

30. Compute the fundamental group of RP 2.

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Projective Geometry