Basics

  1. If $G$ is on $\overline{BY}$ then $3B + 4Y = xG$. What is $x$? What is $BG:GY$?
  2. If $G$ is on $\overline{BY}$ then $7B + xY = 9G$. What is $x$? What is $BG:GY$?
  3. In $\bigtriangleup ABC$, $D$ is the midpoint of $\overline{BC}$ and $E$ is the trisection point of $\overline{AC}$ nearer$A$. Let $G$ = $\overline{BE} \cap \overline{AD}$. Find $AG:GD$ and $BG:GE$.
    Solution: Draw the figure! Assign weight 2 to $A$ and weight 1 to each of $B$ and $C$. Then $2A+1B = 3E$ and $1B+1C=2D$. Note that the center of mass of the system is $2A + 1B + 1C= 3E +1C = 2A + 2D = 4G$. From this we can see that $AG:GD = 2:2 = 1:1$ and $BG:GE = 3:1$.
  4. ( East Bay Mathletes April 1999) In $\bigtriangleup ABC$, $D$ is on $\overline{AB}$ and $E$ is the on $\overline{BC}$. Let $F$ = $\overline{AE} \cap \overline{CD}$. $AD=3$, $DB=2$, $BE=3$ and $EC$=4 Find $EF:FA$ in lowest terms.
  5. Show that the medians of a triangle are concurrent and the point of concurrency divides each median in a ratio of 2:1. (Hint: Assign a weight of 1 to each vertex.) How does this show that the six regions all have the same area?
  6. ( Varignon's Theorem (1654-1722)) If the midpoints of consecutive sides of a quadrilateral are connected, the resulting quadrilateral is a parallelogram. (Hint: Assign weight 1 to each vertex of the original quadrilateral.)
  7. In quadrilateral $ABCD$, $E$, $F$, $G$, and $H$ are the trisection points of $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, and $\overline{DA}$ nearer $A$, $C$, $C$, $A$, respectively. Let $\overline{EG} \cap \overline{FH}=K$. Show that $EFGH$ is a parallelogram.
  8. Generalize the previous problems for $E$, $F$, $G$, and $H$ divide the sides in a ratio of $m:n$.