Problem 5

Let

\triangle OAB

have acute angle

AOB

. Through a point

M\ne O

, draw perpendiculars to

OA

and

OB

, with feet

P

and

Q

. Let

H

be the orthocenter of

\triangle OPQ

. Find the locus of

H

when

M

ranges over (a) the side

AB

; (b) the interior of

\triangle OAB

Step 4 of 4: Read off both loci

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H\\in XY;\\qquad M\\in AB\\Longrightarrow H\\in XY,\\qquad M\\in\\operatorname{int}(OAB)\\Longrightarrow H\\in\\operatorname{int}(OXY).

Detailed analysis

The two perpendiculars through $P$ and $Q$ meet at the orthocenter, so the constructed point on $XY$ is exactly $H$ . As $M$ runs over the side $AB$ , $H$ runs over the segment $XY$ . A line section inside $OAB$ parallel to $AB$ produces a parallel section inside $OXY$ ; varying the section fills the interior of $\triangle OXY$ .