[Math Lair] Pillow-Problems: Problem #24

Math Lair Home > Source Material > Pillow-Problems > Problem #24
Problem #23 | Problem #25
For more information on this collection, see Pillow-Problems by Charles L. Dodgson (Lewis Carroll).

Problem:

24.

If, from the vertices of a triangle ABC, the lines AD, BE,

[diagram for problem 24]
CF be drawn, intersecting at O; find the ratio
DO
DA
in terms of the two ratios
EO
EB
,
FO
FC
.
[5/86

Answer:

24.

DO
DA
+
EO
EB
+
FO
FC
= 1; whence any one can be found in terms of the other two.

Solution:

24.
[diagram for problem 24]

Because
DO
OA
= △
DOC
OAC
= △
DOB
OAB
= △
OBC
OCA + △OAB
; ∴
DO
DA
= △
OBC
ABC
.

Similarly,
EO
EB
= △
OCA
ABC
, and
FO
FC
= △
OAB
ABC
.

Hence
DO
DA
+
EO
EB
+
FO
FC
= 1

Q.E.F.