Difference between revisions of "1965 IMO Problems/Problem 5"

Revision as of 19:01, 30 October 2024

Problem

Consider $\triangle OAB$ with acute angle $AOB$ . Through a point $M \neq O$ perpendiculars are drawn to $OA$ and $OB$ , the feet of which are $P$ and $Q$ respectively. The point of intersection of the altitudes of $\triangle OPQ$ is $H$ . What is the locus of $H$ if $M$ is permitted to range over (a) the side $AB$ , (b) the interior of $\triangle OAB$ ?

Solution

Let $O(0,0),A(a,0),B(b,c)$ . Equation of the line $AB: y=\frac{c}{b-a}(x-a)$ . Point $M \in AB : M(\lambda,\frac{c}{b-a}(\lambda-a))$ . Easy, point $P(\lambda,0)$ . Point $Q = OB \cap MQ$ , $MQ \bot OB$ . Equation of $OB : y=\frac{c}{b}x$ , equation of $MQ : y=-\frac{b}{c}(x-\lambda)+\frac{c}{b-a}(\lambda-a)$ . Solving: $x_{Q}=\frac{1}{b^{2}+c^{2}}\left[b^{2}\lambda+\frac{c^{2}(\lambda-a)b}{b-a}\right]$ . Equation of the first altitude: $x=\frac{1}{b^{2}+c^{2}}\left[b^{2}\lambda+\frac{c^{2}(\lambda-a)b}{b-a}\right] \quad (1)$ . Equation of the second altitude: $y=-\frac{b}{c}(x-\lambda)\quad\quad (2)$ . Eliminating $\lambda$ from (1) and (2): $ac \cdot x + (b^{2}+c^{2}-ab)y=abc$ a line segment $MN , M \in OA , N \in OB$ . Second question: the locus consists in the $\triangle OMN$ .

Solution 2

This solution is a simplified version of the previous solution, it fills in some gaps. and it provides more information. The idea is to just follow the degrees of the expressions and equations in $\lambda, x, y$ involved. If we manage to conclude that the equation for $H$ is an equation of degree $1$ , then we will know that it is a line. We don't need to know the equation explicitly.

Just like in the previous solution, we use analytic (coordinate) geometry, but we don't care how the axes are chosen.

The coordinates of $M$ are expressions of degree $1$ in $\lambda$ .

The equation for $MP$ is an equation of degree $1$ in $x, y$ with constant coefficients for $x, y$ , and whose constant term is an expression of degree $1$ in $\lambda$ .

The coordinates of $P$ (the intersection of $MP$ and $OA$ ) are expressions of degree $1$ in $\lambda$ .

The equation of the perpendicular from $P$ to $OB$ is of degree $1$ in $x, y$ , with constant coefficients for $x, y$ , and whose constant term is an expression of degree $1$ in $\lambda$ . This corresponds to equation (2) in the above solution.

Similarly, the equation of the perpendicular from $Q$ to $OA$ is of degree $1$ in $x, y$ , with constant coefficients for $x, y$ , and whose constant term is an expression of degree $1$ in $\lambda$ . This corresponds to equation (1) in the above solution.

Now, in principle, we would have to solve the system of two equations to obtain the coordinates of $H$ as expressions of $\lambda$ , and then eliminate $\lambda$ to obtain an equation in $x, y$ . Or, as a shortcut, we can eliminate $\lambda$ directly from the two equations. Either way, the result is an equation of degree $1$ in $x, y$ .

This tells us that the locus is on this line. We just need to specify which set of points on this line is the locus.

The previous solution, with a good amount of hand waving, tells us that the solution is "a line segment $B_1A_1, B_1 \in OA, A_1 \in OB$ ". (On top of the hand waving the solution uses the unhappy notation $M$ for $B_1$ and $N$ for $A_1$ , which is bad because $M$ has already been used!) We will do better than that.

Let $A_1$ be the foot of the perpendicular from $A$ to $OB$ , and $B_1$ be the foot of the perpendicular from $B$ to $OA$ . (For this paragraph see the picture shown in Solution 3.) Consider the limit situation when $M = A$ . Then $Q = A_1$ , and $P = A$ . It follows that the intersection $H$ of the perpendiculars from $P$ to $OB$ and $Q$ to $OA$ is $A_1$ . Similarly, the limit situation when $M = B$ yields $H = B_1$ . Now it is reasonable to say that when $M$ moves from $A$ to $B$ , $H$ moves from $A_1$ to $B_1$ . So, the locus is the line segment joining the feet $A_1, B_1$ of the perpendiculars in $\triangle OAB$ from $A, B$ . This answers question (a).

Again with a good amount of hand waving, the previous solution says "the locus consists in the $\triangle OB_1A_1$ ". We justify this by pointing out that if $M$ is inside $\triangle OAB$ , then we can take the triangle $\triangle OA'B'$ , such that $A' \in OA$ , $B' \in OB$ , $A'B'$ going through $M$ and parallel to $AB$ . Then $H$ will be on the corresponding segment $A_1'B_1'$ determined by the feet of the perpendiculars in $\triangle OA'B'$ . Conversely, it is easy to see that any point $H \in \triangle OA_1B_1$ is on a segment $A_1'B_1'$ obtained from a triangle $\triangle OA'B'$ , and $H$ is obtained from a point $M \in A'B'$ . This answers question (b).

(Solution by pf02, October 2024)

Solution 3

TO BE CONTINUED. SAVING MID WAY, SO I DON'T LOSE WORK DONE SO FAR.

@@ Line 23: / Line 23: @@
 This solution is a simplified version of the previous solution,
-and it provides more information.  The idea is to just follow
+it fills in some gaps. and it provides more information.  The
-the degrees of the expressions and equations in <math>\lambda, x, y</math>
+idea is to just follow the degrees of the expressions and
-involved.  If we manage to conclude that the equation for <math>H</math>
+equations in <math>\lambda, x, y</math> involved.  If we manage to conclude
-is an equation of degree <math>1</math>, then we will know that it is a
+that the equation for <math>H</math> is an equation of degree <math>1</math>, then we
-line.  We don't need to know the equation explicitly.
+will know that it is a line.  We don't need to know the equation
+explicitly.
 Just like in the previous solution, we use analytic (coordinate)
@@ Line 44: / Line 45: @@
 <math>1</math> in <math>x, y</math>, with constant coefficients for <math>x, y</math>, and whose
 constant term is an expression of degree <math>1</math> in <math>\lambda</math>.  This
-is equation (2) in the above solution.
+corresponds to equation (2) in the above solution.
 Similarly, the equation of the perpendicular from <math>Q</math> to <math>OA</math>
 is of degree <math>1</math> in <math>x, y</math>, with constant coefficients for
 <math>x, y</math>, and whose constant term is an expression of degree
-<math>1</math> in <math>\lambda</math>.  This is equation (1) in the above solution.
+<math>1</math> in <math>\lambda</math>.  This corresponds to equation (1) in the
+above solution.
+Now, in principle, we would have to solve the system of two
+equations to obtain the coordinates of <math>H</math> as expressions
+of <math>\lambda</math>, and then eliminate <math>\lambda</math> to obtain an
+equation in <math>x, y</math>.  Or, as a shortcut, we can eliminate
+<math>\lambda</math> directly from the two equations.  Either way,
+the result is an equation of degree <math>1</math> in <math>x, y</math>.
+This tells us that the locus is on this line.  We just need to
+specify which set of points on this line is the locus.
+The previous solution, with a good amount of hand waving, tells
+us that the solution is "a line segment
+<math>B_1A_1, B_1 \in OA, A_1 \in OB</math>".  (On top of the hand waving
+the solution uses the unhappy notation <math>M</math> for <math>B_1</math> and <math>N</math>
+for <math>A_1</math>, which is bad because <math>M</math> has already been used!)
+We will do better than that.
+Let <math>A_1</math> be the foot of the perpendicular from <math>A</math> to <math>OB</math>, and
+<math>B_1</math> be the foot of the perpendicular from <math>B</math> to <math>OA</math>.
+(For this paragraph see the picture shown in Solution 3.)
+Consider the limit situation when <math>M = A</math>.  Then <math>Q = A_1</math>, and
+<math>P = A</math>.  It follows that the intersection <math>H</math> of the
+perpendiculars from <math>P</math> to <math>OB</math> and <math>Q</math> to <math>OA</math> is <math>A_1</math>.
+Similarly, the limit situation when <math>M = B</math> yields <math>H = B_1</math>.
+Now it is reasonable to say that when <math>M</math> moves from <math>A</math> to <math>B</math>,
+<math>H</math> moves from <math>A_1</math> to <math>B_1</math>.  So, the locus is the line segment
+joining the feet <math>A_1, B_1</math> of the perpendiculars in
+<math>\triangle OAB</math> from <math>A, B</math>.  This answers question (a).
+Again with a good amount of hand waving, the previous solution
+says "the locus consists in the <math>\triangle OB_1A_1</math>".  We justify
+this by pointing out that if <math>M</math> is inside <math>\triangle OAB</math>, then
+we can take the triangle <math>\triangle OA'B'</math>, such that <math>A' \in OA</math>,
+<math>B' \in OB</math>, <math>A'B'</math> going through <math>M</math> and parallel to <math>AB</math>.  Then
+<math>H</math> will be on the corresponding segment <math>A_1'B_1'</math> determined by
+the feet of the perpendiculars in <math>\triangle OA'B'</math>.  Conversely,
+it is easy to see that any point <math>H \in \triangle OA_1B_1</math> is on
+a segment <math>A_1'B_1'</math> obtained from a triangle <math>\triangle OA'B'</math>,
+and <math>H</math> is obtained from a point <math>M \in A'B'</math>.  This answers
+question (b).
+(Solution by pf02, October 2024)
+== Solution 3 ==

1965 IMO (Problems) • Resources
Preceded by Problem 4	1 • 2 • 3 • 4 • 5 • 6	Followed by Problem 6
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