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 2 of 4: Relate the two side ratios

Previous step Next step

\frac{AM}{MB}=\frac{AP}{PX}=\frac{YH}{HX}.

Detailed analysis

From $MP\\parallel BX$ , triangles in the strip between $AB$ and $XY$ give $AM/MB=AP/PX$ . Draw through $P$ the perpendicular to $OB$ and let it meet $XY$ at $H$ ; it is parallel to $AY$ , so the same parallel-line theorem gives $AP/PX=YH/HX$ .