BAY AREA MATHEMATICAL ADVENTURE

GEOMETRIC PUZZLES AND CONSTRUCTIONS.

THREE DIFFERENT VIEWS OF DESARGUES' THEOREM

BY ZVEZDELINA STANKOVA-FRENKEL, MILLS COLLEGE

Problem 1. (For Everyone to Play With) Three congruent squares with bases $AM,MH$ and $HB$, are put next to each other to form a rectangle $ABCD$ (see Fig.1). Show that ¹

$$\angle AMD+\angle AHD +\angle ABD=90^{\circ}.$$

Problem 2. (For the Die-Hards) Let $ABCDEF$ be a convex hexagon. Let $P,Q,$ and $R$ be the intersections of the lines $AB$ and $EF$, $EF$ and $CD$, $CD$ and $AB$, respectively. Let $S,T,U$ be the intersections of the lines $BC$ and $DE$, $DE$ and $FA$, $FA$ and $BC$, respectively. Show that if $AB/PR=CD/RQ=EF/QP$, then $BC/US=DE/ST=FA/TU$. (Math Olympiad Summer Program'98, Homework Assignment.) ²

Problem 3. (Desargues' Theorem) $\triangle ABC$ and $\triangle A_1B_1C_1$ are positioned in such a way that lines $AA_1$, $BB_1$, and $CC_1$ intersect in a point $O$. If lines $AB$ and $A_1B_1$, $AC$ and $A_1C_1$, $BC$ and $B_1C_1$ are pairwise not parallel, prove that their points of intersection, $L$, $M$ and $N$, are collinear.

Problem 4. (Pascal's Theorem) If the hexagon $ABCDEF$ is cyclic and its opposite sides, $AB$ and $DE$, $BC$ and $EF$, $CD$ and $FA$, are pairwise not parallel, prove that their three points of intersection, $X$, $Y$ and $Z$, are collinear.

Problem 5. (Brianchon's Theorem) If the hexagon $ABCDEF$ is circumscribed around a circle, prove that its three diagonals $AD,BE$ and $CF$ are concurrent.

Note: ``We say that several points are collinear if they lie on a line. Similarly, several points are concyclic if they lie on a circle; an inscribed (cyclic) polygon has its vertices lying on a circle. If three distinct points $A$, $B$ and $C$ are collinear, then the directed ratio $\overline{AB}/\overline{CB}$ is the ratio of the lengths of segments $AB$ and $CB$, taken with a sign ``$+$'' if the segments have the same direction (i.e. $B$ is not between $A$ and $C$), and with a sign ``$-$'' if the segments have opposite directions (i.e. $B$ is between $A$ and $C$). Several objects (lines, circles, etc.) are concurrent if they all intersect in some point.

Problem 6. (Menelaus' Theorem) Let $A_1,B_1$ and $C_1$ be three points on the sides $BC,CA$ and $AB$ of $\triangle ABC$. Then they are collinear if and only if

$$\frac{\overline{AB_1}}{\overline{CB_1}} \cdot\frac{\overline{CA_1}}{\overline{BA_1}} \cdot\frac{\overline{BC_1}}{\overline{AC_1}}=1.$$