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Source: 32-th Vietnamese Mathematical Olympiad 1994

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#1 h Feb 24, 2007, 6:07 PM • 3 Y

Y by Adventure10, Mango247, and 1 other user

$S$ is a sphere center $O. G$ and $G'$ are two perpendicular great circles on $S$ . Take $A, B, C$ on $G$ and $D$ on $G'$ such that the altitudes of the tetrahedron $ABCD$ intersect at a point. Find the locus of the intersection.

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#2 h Feb 26, 2011, 4:25 AM • 2 Y

Y by Adventure10, Mango247

Let $O$ be the origin and the $z$ -axis is perpendicular to $G$ . Notice that
$\vec {A}\cdot \vec {A}=\vec {B}\cdot \vec {B}=\vec {C}\cdot \vec {C}=\vec {D}\cdot \vec {D}$
and that the orthocenter $H$ of $\triangle ABC$ is
$\vec{A}+\vec{B}+\vec{C}$
It is well-know that if $ABCD$ has an orthocenter then the foot of $D$ to $\triangle ABC$ coincides with the orthocenter of $\triangle ABC$ . Thus,
$\vec{D}=\vec{A}+\vec{B}+\vec{C}+\vec{Z}$
where $\vec{Z}$ is a vector that is parallel to the $z$ -axis. We also have that the orthocenter, $K$ of $ABCD$ is
$\vec{A}+\vec{B}+\vec{C}+\vec{Z'}$
wher $\vec{Z'}$ is another vector that is parallel to the $z$ -axis. By definintion of the orthocenter, $\vec{AK}$ is perpendicular to $\triangle BCD$ so $\vec{AK}$ is perpendicular to $\vec{BD}$ . Thus,
$\begin{align*} 0&=\vec{AK}\cdot \vec{BD}\\ &=(\vec K-\vec A)\cdot (\vec D-\vec B)\\ &=(\vec B+\vec C+\vec {Z'})\cdot (\vec A+\vec C+\vec Z)\\ &=(\vec B+\vec C)\cdot (\vec A+\vec C)+\vec {Z'}\cdot \vec Z\\ &=\frac{1}{2}(|\vec{A}+\vec{B}+\vec{C}|^2-|\vec{A}|^2)+|\vec {Z'}||\vec Z|\\ |\vec {Z'}|&=\frac{\frac{1}{2}(|\vec Z|^2)}{\vec Z}\\ &=\frac{1}{2}{|\vec Z|}\end{align*}$
Thus, the locus of $K$ is a subset the curve formed by reducing the $z$ -coordinate of each possible $D$ in half. Since we can construct for each $D$ , a triangle $ABC$ such that the foot of $D$ to $\triangle ABC$ is the orthocenter of $ABC$ , and that is can be proven that $ABCD$ does have an orthocenter, the locus of $K$ is this curve.

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#3 h Dec 26, 2011, 6:28 AM • 2 Y

Y by Adventure10, Mango247

The locus is a concentric sphere with the same center O and radius R/4 where R is the radius of the great circles.

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