PillowProblems
: Problem #24
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Problem #24
Problem #23

Problem #25
For more information on this collection, see
PillowProblems
by Charles L. Dodgson (Lewis Carroll)
.
Problem:
24.
If, from the vertices of a triangle
ABC
, the lines
AD
,
BE
,
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.
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.