Topics in Projective Geometry
Cross-Ratios
1. Denition: Let A, B, C three collinear points in this order. Then there exists exactly AB AD one other point D AC so that = . In this case one says that B and D are BC CD harmonic conjugates with respect to A and C. 2. Denition: Let A, B, C, D be four collinear points in this order. Then one denes the AD AB AB AD cross-ratio (A, B, C, D) = : = BC : CD .
BC CD
3. Pappuss Theorem: Let (Oxi for i = 1, . . . , 4 be four rays (forming a total angle of < 180) and Ai , Bi (Oxi so that A1 , A2 , A3 , A4 and B1 , B2 , B3 , B4 are collinear respectively. Then (A1 , A2 , A3 , A4 ) = (B1 , B2 , B3 , B4 ).
Polar Transformations
1. Let C be a circle and A a point outside it. Let l be a line containing A, l C = {M, N }, and let B be the harmonic conjugate of A with respect to M and N. Find the locus of B, and denote it by pA . This is called the polar of A with respect to C. 2. M pA M O2 M A2 = 2R2 OA2 , where O is the center of C and R is its radius. 3. Prove that B pA A pB . 4. Prove that one can dene the polar for any point other than O. 5. Let l be a line not containing O. Then there is a point A so that l = pA . This point is called the pole of l. 6. Prove that A, B, C are collinear pA , pB , pC are concurrent. 7. Dual tranformation. Let C be a projective conguration of points and lines (projective means that the only things that count are collinearities and concurrences). Choose a circle whose center is not among the points of C. Then take the polar of each point and the pole of each line in C. Then in the new conguration, all previous collinearities become concurrences and vice-versa. This is a very ecient method of simplifying problems.
Problems 1. Let C be a circle and P a point outside it. A line l through P intersects C in M and N. The tangents through M and N to C intersect at Q. Prove that Q describes a line. 1
2. Let ABCD be a cyclic quadrilateral with circumcircle C. Let AB {V }, AC BD = {T }. Prove that U T = pV .
CD = {U }, BC
AD =
3. Let ABC be an acute triangle. The interior bisectors of B and C meet the opposite sides in L, M respectively. Prove that there is a point K (BC) such that KLM is equilateral A = 60. (Romanian Selection Test, 1999)
Projective Transformation
The set of all points of intersections of parallel lines is called the line at innity of the Euclidian plane. Let be the projective plane, i.e. the union of the Euclidian plane with the line at innity. Let l and O \l. Consider a point V outside the projective plane. Take d a line containing O. Let a plane through d, parallel to the plane determined by l and V . . For each point X let Y = f (X) = V X . Then this is called the projective transformation that sends l to innity. The reason for this is quite obvious, i.e. for any X l then f (Y ) will be on the line to innity of . In this case one just says that one throws l to innity This method is very ecient, because the following properties imply that any two lines intersecting on l will be transformed into two parallel lines. 1. Prove that f takes lines to lines. 2. Prove that f invaries cross-ratios. Problems 1. Let ABC be a triangle and M, N, P be three points on sides BC, CA, AB respectively. Let D, E, F be the harmonic conjugates of M, N, P with respect to the endpoints of the sides on which they are. Prove that D, E, F are collinear AM, BN, CP are concurrent. 2. Pappuss Theorem (come on, how many are there? ). Let a b = {P } be two intersecting lines. Let Ai a, Bi b for i = 1, 2, 3. Prove that Ai Bi intersect (P, A1 , A2 , A3 ) = (P, B1 , B2 , B3 ). 3. Guess what? Pappus Theorem, again... Let d1 , d2 be two lines and A1 , A2 , A3 d1 and B1 , B2 , B3 d2 distinct points. If {Ui } = Aj Bk Ak Bj for any permutation {i, j, k} = {1, 2, 3} then U1 , U2 , U3 are collinear. 4. Let U, V BC in triangle ABC. A line parallel to AC intersects AB, AU, AV in P, Q, R respectively. Find P Q . QR
5. Let ABCD be a convex quadrilateral with O = AD BC, T = AC BD. We consider points P and Q on OT. Let be U and V the intersection points of the diagonals of AQPD and BQPC respectively. Let M and N be two points on (AD) and (BC). Prove that OT,MU si NV are concurrent if and only if AB,DC and MN are concurrent.(my problem, i.e.) 6. (My favorite...) Let V A1 . . . An be a pyramid and Bk (V Ak ). Let {Ti,j } = Ai Bj Aj Bi . Prove that if at least 1 + n1 of the points Ti,j are coplanar then all points Ti,j are 2 coplanar. (County olympiad, Arad 1999. Proposed by Iani Mirciov ) Compiled by Andrei Jorza
