Angle Bisectors, Nonconcurrency, Mass Points in Space

  1. In $\bigtriangleup ABC$, $AB = 8$, $BC = 6$ and $CA = 7$. Let $P$ be the incenter of the triangle and $D$, $E$, $F$ be the intersection points of the angle bisectors in side $\overline{BC}$, $\overline{CA}$ and $\overline{AB}$, respectively. Find $AP:PD$, $BP:PE$, and $CP:PF$. (Hint: Assign weight 6 to $A$, weight 7 to $B$ and weight 8 to $C$.)
  2. Solve the previous problem using $AB = c$, $BC = a$ and $CA = b$.
  3. Use the previous problem to prove, as assumed in the previous two problems, that the angle bisectors of the angles of a triangle are concurrent.
  4. In $\bigtriangleup ABC$, $D$, $E$, and $F$ are the trisection points of $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$ nearer $A$,$B$,$C$, respectively. Let $\overline{BF} \cap \overline{AE} = J$. Show that $BJ:JF = 3:4$ and $AJ:JE = 6:1$.
  5. In the previous problem, let $\overline{CD} \cap \overline{AE}=K$ and $\overline{CD} \cap \overline{BF}=L$. Use the previous problem to show that $DK:KL:LC=1:3:3=EJ:JK:KA =FL:LJ:JB.$
  6. Let $ABCD$ be a tetrahedron (triangular pyramid). Assume the same definitions and properties of addition of mass points in space as for in the plane. Assign weights of 1 to each of the vertices. Let $G$ be the point in $\bigtriangleup ABC$ such that $1A + 1B +1C = 3G$. Then $G$ is the center of mass for $\bigtriangleup ABC$. Let $F$ be the point on $\overline{DG}$ such that $1D+3G= F$. $F$ is the center of mass of the tetrahedron. What is the ratio of $DF$ to $FG$?
  7. Show that the four segments from the vertices to centroids of the opposite faces are concurrent at the point $F$ of the previous problem.
  8. In tetrahedron $ABCD$, let $E$ be in $\overline{AB}$ such that $AE:EB=1:2$, let $H$ be in $\overline{BC}$ such that $BH:HC=1:2$, and let $\overline{AH} \cap \overline{CE}=K$. Let $M$ be the midpoint of $\overline{DK}$ and let ray $HM$ intersect $\overline{AD}$ in $L$. Show that $AL:LD = 7:4$.
  9. Show that the three segments joining the midpoints of opposite edges of a tetrahedron bisect each other. (Opposite edges have no vertex in common.)
  10. Let $P-ABCD$ be a pyramid on convex base $ABCD$ with $E$, $F$, $G$, and $H$ the midpoints of $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, and $\overline{DA}$. Let $E^\prime$, $F^\prime$, $G^\prime$, and $H^\prime$, be the respective centroids of $\bigtriangleup$'s $PCD,PDA, PAB$, and $PBC$. Show that $\overline{EE^\prime},\overline{FF^\prime},\overline{GG^\prime}, \overline{HH^\prime}$ are concurrent in a point $K$ which divides each of the latter segments in a ratio of 2:3.