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- Two triangles
*A*_{1}*B*_{1}*C*_{1}and*A*_{2}*B*_{2}*C*_{2}in**R**^{3}are in perspective from*O*if*A*_{1}*A*_{2},*B*_{1}*B*_{2}and*C*_{1}*C*_{2}meet at*O*. The intersection of the planes*A*_{1}*B*_{1}*C*_{1}and*A*_{2}*B*_{2}*C*_{2}is called the*axis of perspective*.

Prove that for two such triangles the four axes of perspective of the pairs*A*_{1}*B*_{1}*C*_{1},*A*_{2}*B*_{2}*C*_{2}:*A*_{2}*B*_{1}*C*_{1},*A*_{1}*B*_{2}*C*_{2}:*A*_{1}*B*_{2}*C*_{1},*A*_{2}*B*_{1}*C*_{2}:*A*_{1}*B*_{1}*C*_{2},*A*_{2}*B*_{2}*C*_{1}are coplanar.

[Project the axis of the pair*A*_{1}*B*_{1}*C*_{1},*A*_{2}*B*_{2}*C*_{2}and the point*O*to the plane at infinity in**R***P*^{3}and then look at what this does to the picture.] - Show that the elements of
*PGL*(2,**R**) which map a line to itself form a subgroup isomorphic to the Affine group*A*(**R**^{2}).

Deduce that one may recover Affine Geometry from Projective Geometry by considering only those maps which take the line at infinity to itself. - Deduce a result from Pascal's Mystic Hexagram Theorem by making consecutive vertices coincide.

Use Brianchon's theorem to deduce a property of a triangle which circumscribes a conic. (i.e. the three sides of the triangle are tangents to the conic.)

In the case when the conic is a circle, give an independent proof of this. - Given five points on a conic, use Pascal's theorem to construct any number of other points on the conic.
- A circle in the elliptic plane is a set {
*y*|*d*(*x*,*y*) =*r*} for a fixed*x***R***P*^{2}and*r*> 0.

By drawing the corresponding curves on the 2-sphere*S*^{2}find two circles in the elliptic plane which intersect in four points. - If an elliptic triangle has angles 0 < then prove that + < p + .

For which integers*p*,*q*,*r*2 is there an elliptic triangle with angles*p*/*p*,*p*/*q*,*p*/*r*? - In what range do the angles of a regular
*p*-gon in the elliptic plane lie?

Which regular*p*-gons can be used to tile the elliptic plane?

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