More Problems

  1. ( AHSME 1964 #35) The sides of a triangle are of lengths 13, 14, and 15. The altitudes of the triangle meet at point $H$. If $\overline{AD}$ is the altitude to the side of length 14, what is the ratio $HD:HA$?
  2. ( AHSME 1965 #37) Point $E$ is selected on side $AB$ of triangle $ABC$ in such a way that $AE:EB=1:3$ and point $D$ is selected on side $BC$ so that $CD:DB = 1:2$. The point of intersection of $AD$ and $CE$ is $F$. Find $\frac{EF}{FC}+\frac{AF}{FD}$.
  3. ( AHSME 1975 #28) In triangle $ABC$, $M$ is the midpoint of side $BC$, $AB=12$ and $AC=16$. Points $E$ and $F$ are taken on $AC$ and $AB$, respectively, and lines $EF$ and $AM$ intersect at $G$. If $AE=2AF$ then find $EG/GF$.
  4. ( AHSME 1980 #21) In triangle $ABC$, $\angle CBA=72^\circ$, $E$ is the midpoint of side $AC$ and $D$ is a point on side $BC$ such that $2BD=DC$; $\overline{AD}$ and $\overline{BE}$ intersect at $F$. Find the ratio of the area of triangle $BDF$ to the area of quadrilateral $FDCE$.
  5. ( NYSML S75 #27) In $\bigtriangleup ABC$, $C^\prime$ is on $\overline{AB}$ such that $AC^\prime :C^\prime B=1:2$, and $B^\prime$ is on $\overline{AC}$ such that $AB^\prime :B^\prime C = 3:4$. If $\overline{BB^\prime} \cap \overline{CC^\prime}= P$ and if $A^\prime$ is the intersection of ray $AP$ and $\overline{BC}$ then find $AP:PA^\prime$.
  6. ( NYSML F75 #12) In $\bigtriangleup ABC$, $D$ is on $\overline{AB}$ and $E$ is on $\overline{BC}$. Let $\overline{CD} \cap \overline{AE}=K$ and let ray $BK \cap \overline{AC} =F$. If $AK:KE=3:2$ and $BK:KF=4:1$, then find $CK:KD$.
  7. ( NYSML F76 #13) In $\bigtriangleup ABC$, $D$ is on $\overline{AB}$ such that $AD:DB=3:2$ and $E$ is on $\overline{BC}$ such that $BE:EC=3:2$. If ray $DE$ and ray $AC$ intersect at $F$, then find $DE:EF$.
  8. ( NYSML S77 #1) In a triangle, segments are drawn from one vertex to the trisection points of the opposite side . A median drawn from a second vertex is divided , by these segments, in the continued ratio $x:y:z$. If $x \ge y \ge z$ then find $x:y:z$.
  9. ( NYSML S77 #22) A circle is inscribed in a 3-4-5 triangle. A segment is drawn from the smaller acute angle to the point of tangency on the opposite side. This segment is divided in the ratio $p:q$ by the segment drawn from the larger acute angle to the point of tangency on its opposite side. If $p>q$ then find $p:q$.
  10. ( NYSML S78 #25) In $\bigtriangleup ABC$, $\angle A =45^\circ$ and $\angle C =30^\circ$. If altitude $\overline{BH}$ intersects median $\overline{AM}$ at $P$, then $AP:PM=1:k$. Find $k$.
  11. ( NYSML F80 #13) In $\bigtriangleup ABC$, $D$ is the midpoint $\overline{BC}$ and $E$ is the midpoint of $\overline{AD}$. If ray $BE$ and intersects $\overline{AC}$ at $F$. Find the value of $\frac{FE}{EB} +\frac{AF}{FC}$.
  12. ( ARML 1989 T4) In $\bigtriangleup ABC$, angle bisectors $\overline{AD}$ and $\overline{BE}$ intersect at $P$. If $a =3$, $b=5$, $c=7$, $BP=x$, and $PE =y$, compute the ratio $x:y$, where $x$ and $y$ are relatively prime integers.
  13. ( ARML 1992 I8) In $\bigtriangleup ABC$, points $D$ and $E$ are on $\overline{AB}$ and $\overline{AC}$, respectively. The angle bisector of $\angle A$ intersects $\overline{DE}$ at $F$ and $\overline{BC}$ at $T$. If $AD=1,DB=3,AE=2$, and $EC = 4$, compute the ratio $AF:AT$
  14. In the previous problem, if $AD=a, AB=b,AE=c$ and $AC=d$ then show that $\frac{AF}{AT}= \frac{ac(b+d)} {bd(a+c)}$.
  15. ( AIME 1985 #6) In triangle $ABC$, cevians $\overline{AD}$, $\overline{BE}$ and $\overline{CF}$ intersect at point $P$. The areas of triangles $PAF,PFB,PBD$ and $PCE$ are 40,30,35 and 84, respectively. Find the area of triangle $ABC$.
  16. ( AIME 1988 #12) Let $P$ be an interior point of triangle $ABC$ and extend lines from the vertices through $P$ to the opposite sides. Let $AP=a$, $BP=b$, $CP=c$ and the extensions from $P$ to the opposite sides all have length $d$. If $a+b+c = 43$ and $d=3$ then find $abc$.
  17. ( AIME 1989 #15) Point $P$ is inside triangle $ABC$. Line segments $\overline{APD}$, $\overline{BPE}$, and $\overline{CPF}$ are drawn with $D$ on $\overline{BC}$, $E$ on $\overline{CA}$, and $F$ on $\overline{AB}$. Given that $AP=6$, $BP=9$, $PD =6$, $PE=3$, and $CF=20$, find the area of triangle $ABC$.
  18. ( AIME 1992 #14) In triangle ABC, $A^\prime$, $B^\prime$, and $C^\prime$ are on sides $\overline{BC}$, $\overline{AC}$, $\overline{AB}$, respectively. Given that $\overline{AA^\prime}$, $\overline{BB^\prime}$, and $\overline{CC^\prime}$ are concurrent at the point $O$, and that $ \frac{AO}{OA^\prime} + \frac{BO}{OB^\prime} + \frac{CO}{OC^\prime}=92$, find the value of $\frac{AO}{OA^\prime} \cdot \frac{BO}{OB^\prime} \cdot \frac{CO}{OC^\prime}$.
  19. (Larson [ 14] problem 8.3.4) In triangle $ABC$, let $D$ and $E$ be the trisection points of $BC$ with $D$ between $B$ and $E$. Let $F$ be the midpoint of $\overline{AC}$, and let $G$ be the midpoint of $\overline{AB}$. Let $H$ be the intersection of $\overline{EG}$ and $\overline{DF}$. Find the ratio $EH:HG$.
  20. Use nonconcurrency problems #4 and #5 to show that the triangle $\bigtriangleup JKL$ is one-seventh the area of $\bigtriangleup ABC$. Generalize the problem using points which divide the sides in a ratio of $1:n$ to show the ratio of the areas is $(1-n)^3:(1-n^3)$. This can be generalized even further using different ratios on each side. It is known as Routh's Theorem. See [ 15] [ 16] and [ 17].