## Problems in Projective Geometry

The following list of problems is aimed to those who want to practice projective geometry. They cover topics such as cross ration, harmonic conjugates, poles and polars, and theorems of Desargue, Pappus, Pascal, Brianchon, and Brocard.

Problem 1 Given a quadrilateral $$ABCD$$, let $$P=AB\cap CD$$, $$Q=AD\cap BC$$, $$R=AC\cap PQ$$, $$S=BD\cap PQ$$. Prove that $$\mathcal H(P,Q;R,S)$$.

Problem 2 Given a triangle $$ABC$$ and a point $$M$$ on $$BC$$, let $$N$$ be the point of the line $$BC$$ such that $$\angle MAN=90^{\circ}$$. Prove that $$\mathcal H(B,C;M,N)$$ if and only if $$AM$$ is the bisector of the angle $$\angle BAC$$.

Problem 3 Let $$A$$ and $$B$$ be two points and let $$C$$ be the point of the line $$AB$$. Using just a ruler find a point $$D$$ on the line $$AB$$ such that $$\mathcal H(A,B;C,D)$$.

Problem 4 Let $$A$$, $$B$$, $$C$$ be the diagonal points of the quadrilateral $$PQRS$$, or equivalently $$A=PQ\cap RS$$, $$B=QR\cap SP$$, $$C=PR\cap QS$$. If only the points $$A$$, $$B$$, $$C$$, $$S$$, are given using just a ruler construct the points $$P$$, $$Q$$, $$R$$.

Problem 5 Assume that the incircle of $$\triangle ABC$$ touches the sides $$BC$$, $$AC$$, and $$AB$$ at $$D$$, $$E$$, and $$F$$. Let $$M$$ be the point such that the circle $$k_1$$ incscribed in $$\triangle BCM$$ touches $$BC$$ at $$D$$, and the sides $$BM$$ and $$CM$$ at $$P$$ and $$Q$$. Prove that the lines $$EF$$, $$PQ$$, $$BC$$ are concurrent.

Problem 6 Given a triangle $$ABC$$, let $$D$$ and $$E$$ be the points on $$BC$$ such that $$BD=DE=EC$$. The line $$p$$ intersects $$AB$$, $$AD$$, $$AE$$, $$AC$$ at $$K$$, $$L$$, $$M$$, $$N$$, respectively. Prove that $$KN\geq 3 LM$$.

Problem 7 The point $$M_1$$ belongs to the side $$AB$$ of the quadrilateral $$ABCD$$. Let $$M_2$$ be the projection of $$M_1$$ to the line $$BC$$ from $$D$$, $$M_3$$ projection of $$M_2$$ to $$CD$$ from $$A$$, $$M_4$$ projection of $$M_3$$ to $$DA$$ from $$B$$, $$M_5$$ projection of $$M_4$$ to $$AB$$ from $$C$$, etc. Prove that $$M_{13}=M_1$$.

Problem 8 (Butterfly Theorem) Points $$M$$ and $$N$$ belong to the circle $$k$$. Let $$P$$ be the midpoint of the chord $$MN$$, and let $$AB$$ and $$CD$$ ($$A$$ and $$C$$ are on the same side of $$MN$$) be arbitrary chords of $$k$$ passing through $$P$$. Prove that lines $$AD$$ and $$BC$$ intersect $$MN$$ at points that are equidistant from $$P$$.

Problem 9 Given a triangle $$ABC$$, let $$D$$ and $$E$$ be the points of the sides $$AB$$ and $$AC$$ respectively such that $$DE\| BC$$. Let $$P$$ be an interior point of the triangle $$ADE$$. Assume that the lines $$BP$$ and $$CP$$ intersect $$DE$$ at $$F$$ and $$G$$ respectively. The circumcircles of $$\triangle PDG$$ and $$\triangle PFE$$ intersect at $$P$$ and $$Q$$. Prove that the points $$A$$, $$P$$, and $$Q$$ are collinear.

Problem 10 (IMO 1997 shortlist) Let $$A_1A_2A_3$$ be a non-isosceles triangle with the incenter $$I$$. Let $$C_i$$, $$i=1$$, $$2$$, $$3$$, be the smaller circle through $$I$$ tangent to both $$A_iA_{i+1}$$ and $$A_iA_{i+2}$$ (summation of indeces is done modulus 3). Let $$B_i$$, $$i=1$$, $$2$$, $$3$$, be the other intersection point of $$C_{i+1}$$ and $$C_{i+2}$$. Prove that the circumcenters of the triangles $$A_1B_1I$$, $$A_2B_2I$$, $$A_3B_3I$$ are collinear.

Problem 11 Given a triangle $$ABC$$ and a point $$T$$, let $$P$$ and $$Q$$ be the feet of perpendiculars from $$T$$ to the lines $$AB$$ and $$AC$$, respectively. Let $$R$$ and $$S$$ be the feet of perpendiculars from $$A$$ to $$TC$$ and $$TB$$, respectively. Prove that the intersection of $$PR$$ and $$QS$$ belongs to $$BC$$.

Problem 12 Given a triangle $$ABC$$ and a point $$M$$, a line passing through $$M$$ intersects $$AB$$, $$BC$$, and $$CA$$ at $$C_1$$, $$A_1$$, and $$B_1$$, respectively. The lines $$AM$$, $$BM$$, and $$CM$$ intersect the circumcircle of $$\triangle ABC$$ repsectively at $$A_2$$, $$B_2$$, and $$C_2$$. Prove that the lines $$A_1A_2$$, $$B_1B_2$$, and $$C_1C_2$$ intersect in a point that belongs to the circumcircle of $$\triangle ABC$$.

Problem 13 Let $$P$$ and $$Q$$ isogonaly conjugated points and assume that $$\triangle P_1P_2P_3$$ and $$\triangle Q_1Q_2Q_3$$ are their pedal triangles, respectively.

Problem 14 If the points $$A$$ and $$M$$ are conjugated with respect to $$k$$, then the circle with diameter $$AM$$ is orthogonal to $$k$$.

Problem 15 From a point $$A$$ in the exterior of a circle $$k$$ two tangents $$AM$$ and $$AN$$ are drawn. Assume that $$K$$ and $$L$$ are two points of $$k$$ such that $$A, K, L$$ are colinear. Prove that $$MN$$ bisects the segment $$PQ$$.

Problem 16 The point isogonaly conjugated to the centroid is called the Lemuan point. The lines connected the vertices with the Lemuan point are called symmedians. Assume that the tangents from $$B$$ and $$C$$ to the circumcircle $$\Gamma$$ of $$\triangle ABC$$ intersect at the point $$P$$. Prove that $$AP$$ is a symmedian of $$\triangle ABC$$.

Problem 17 Given a triangle $$ABC$$, assume that the incircle touches the sides $$BC$$, $$CA$$, $$AB$$ at the points $$M$$, $$N$$, $$P$$, respectively. Prove that $$AM$$, $$BN$$, and $$CP$$ intersect in a point.

Problem 18 Let $$ABCD$$ be a quadrilateral circumscribed about a circle. Let $$M$$, $$N$$, $$P$$, and $$Q$$ be the points of tangency of the incircle with the sides $$AB$$, $$BC$$, $$CD$$, and $$DA$$ respectively. Prove that the lines $$AC$$, $$BD$$, $$MP$$, and $$NQ$$ intersect in a point.

Problem 19 Let $$ABCD$$ be a cyclic quadrilateral whose diagonals $$AC$$ and $$BD$$ intersect at $$O$$; extensions of the sides $$AB$$ and $$CD$$ at $$E$$; the tangents to the circumcircle from $$A$$ and $$D$$ at $$K$$; and the tangents to the circumcircle at $$B$$ and $$C$$ at $$L$$. Prove that the points $$E$$, $$K$$, $$O$$, and $$L$$ lie on a line.

Problem 20 Let $$ABCD$$ be a cyclic quadrilateral. The lines $$AB$$ and $$CD$$ intersect at the point $$E$$, and the diagonals $$AC$$ and $$BD$$ at the point $$F$$. The circumcircle of the triangles $$\triangle AFD$$ and $$\triangle BFC$$ intersect again at $$H$$. Prove that $$\angle EHF=90^{\circ}$$.