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Source: Greek national M.O. 2003, Final Round,problem 3

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#1 h Nov 15, 2011, 10:57 PM • 1 Y

Y by Adventure10

Given are a circle $\mathcal{C}$ with center $K$ and radius $r,$ point $A$ on the circle and point $R$ in its exterior. Consider a variable line $e$ through $R$ that intersects the circle at two points $B$ and $C.$ Let $H$ be the orthocenter of triangle $ABC.$

Show that there is a unique point $T$ in the plane of circle $\mathcal{C}$ such that the sum $HA^2 + HT^2$ remains constant (as $e$ varies.)

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#2 h Nov 16, 2011, 6:15 AM • 2 Y

Y by Adventure10, Mango247

$AH \perp BC$ cuts the line $BC$ at $D$ and the circle $\mathcal C$ again at $X$ . Because of the right angle $\angle ADR$ , $D$ is on a circle $\mathcal D$ on diameter $AR$ .
Powers of $H$ to the circles $\mathcal C, \mathcal D$ are in fixed ratio $\frac{\overline{HA} \cdot \overline{HX}}{\overline{HA} \cdot \overline{HD}} = \frac{\overline{HX}}{\overline{HD}} = 2$ $\Longrightarrow$ $H$ is on a circle $\mathcal H$ coaxal with the circles $\mathcal C, \mathcal D$ $\Longrightarrow$ $A \in \mathcal H$ .
See http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=121062&p=687577#p687577 for the theorem.
Let $AT$ be diameter of the circle $\mathcal H$ $\Longrightarrow$ $HA^2 + HT^2 = AT^2 = \text{const}$ .

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#3 h Dec 3, 2011, 6:11 AM • 2 Y

Y by Adventure10, Mango247

it suffices to prove that H lies on a certain fixed circle.notice that the midpoint of $BC$ ( $M$ ) lies on a circle with dianeter $KR$ .consider the product of two transpositions:
$H(A,\frac{2}{3});H(O,3)$
in this respect M turns into H.

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