Difference between revisions of "1965 IMO Problems/Problem 5"
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a line segment <math>MN , M \in OA , N \in OB</math>. | a line segment <math>MN , M \in OA , N \in OB</math>. | ||
Second question: the locus consists in the <math>\triangle OMN</math>. | Second question: the locus consists in the <math>\triangle OMN</math>. | ||
+ | |||
+ | == See Also == | ||
+ | {{IMO box|year=1965|num-b=4|num-a=6}} |
Revision as of 12:51, 29 January 2021
Problem
Consider with acute angle
. Through a point
perpendiculars are drawn to
and
, the feet of which are
and
respectively. The point of intersection of the altitudes of
is
. What is the locus of
if
is permitted to range over (a) the side
, (b) the interior of
?
Solution
This problem needs a solution. If you have a solution for it, please help us out by adding it.
Let .
Equation of the line
.
Point
.
Easy, point
.
Point
,
.
Equation of
, equation of
.
Solving:
.
Equation of the first altitude:
.
Equation of the second altitude:
.
Eliminating
from (1) and (2):
a line segment
.
Second question: the locus consists in the
.
See Also
1965 IMO (Problems) • Resources | ||
Preceded by Problem 4 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 6 |
All IMO Problems and Solutions |