### PROJECTIVE GEOMETRY COURSE

Extra problems

1. K is a non-degenerated conic through two points A and B, and the line l intersects K in two other points C and D. For X on l, let SX be the "second" intersection point of AX and K, and TX the intersection point of l and the polar line of BSX .l.
a) Prove that the mapping ψ : XTX is a projectivity.
b) Suppose K has equation x12 + x22 = x32 ; A = λ(1,0,1), B = λ(-1,0,1), C = λ(0,1,1), D = λ(0,-1,1). Prove that in this special case ψ is identity.
c) Make a sketch from which it becomes evident that ψ isn't identity in all cases.
d) Prove that ψ is hyperbolic if it's not identity.
e) Suppose we have on K given points A, B, E, F, G, not on l, whilst l is entirely given. Show how we can construct the image of AE . l using only a ruler.
f) Show how we can construct C and D, using ruler and compasses only, starting from the situation given in e), after we have constructed the images of AE . l, AF . l and AG . l.
g) Take the situation given in b), except B: λ(-(1/2)√2, (1/2)√2, 1) instead of B: λ(-1, 0, 1). Determine the projective coordinates of the image of AB . l, and construct the matrix on the basis ((0,1,0),(0,0,1)) of a linear transformation of {x1=0} that induces ψ.

2. Given a non-degenerated conic K and two lines l and m, whilst the pole of m doesn't lie on l. We map each point X on l to π(X) = x . m, where x is the polar line of X with respect to K.
Prove that π is a projectivity from l onto m, and that the Pappus line of π goes through the poles L of l and M of m.

3. Given five freely situated points and a line that doesn't go through any of these five points. Construct all intersection points (if there are any) of the line and the conic through the five points.
(Hint: consider the projectivity x . l → φ(x) . l, where φ is a projectivity between pencils of lines that generates the conic.)

4. Given four proper points on a parabola and the direction of its axis. Construct its top.

5. Given two parallel lines l and m and four points A, B, C, D, whilst A and B don't lie on l and C and D don't lie on m.
Construct a point X on l and a point Y on m such that AX is parallel to CY and BX is parallel to DY.
(Hint: first choose a suitable projectivity from l onto itself.)

6. A and B are two points of a conic K. The tangents to K in A and B intersect each other in S. An arbitrary line through S intersects K in C and D. The tangents SA and SB intersect the tangent in C in the intersection points E and F; the lines DA and DB intersect this last tangent in G and H. Finally, L is the intersection point of AB and this tangent in C.
Show that L and C separate harmonically with both the pairs {E, F} and {G, H}. Prove that the tangent to K in D is also tangent to the conic through A, B, D, E and F.

7. In a plane the points A, B, P and Q are given. A and B don't lie on PQ. We consider the translation φ over vector q-p.
What is the geometric locus of the intersection points AX . Bφ(X) with X on PQ?

8. Given the parabola x2x3 = x12. Determine the projective coordinates of the polar line of the top, the polar line of the point at infinity on the axis, and the pole of the axis.

9. Of a conic K five points A, B, C, S, T are given. A conic K ' goes through A, B, C, U, V. Construct, if possible, an intersection point of K and K ' that's not A, B or C. Which cases do occur?
(Hint: consider a suitable projectivity φ between pencils of lines that generates K and a suitable projectivity ψ between pencils of lines that generates K '.)

10. A variable conic K has four fixed tangents a, b, c, d. From a fixed point A on a we draw the other tangent α to K, and from a fixed point D on d the other tangent δ to K. Prove that α . δ runs through a fixed straight line.
(Hint: Brianchon.)

11. Through a point O we draw four lines that intersect a conic K in (respectively) A and A1, B and B1, C and C1, D and D1. AB and CD intersect each other in L, A1B1 and C1D1 intersect each other in M.
Prove that O, L and M are collinear.

12. Given the triangle ABC and the line l. Suppose a variable conic has l and the sides of the triangle among its tangents. The contact points on BC, AC and AB are A1, B1 and C1, respectievely. Prove:
a) AA1, BB1 and CC1 go through one and the same point P.
b) The pencils (A1) and (B1) of lines are perspective.
c) The variable point P lies on a fixed conic through A, B and C.