2 lines intersect on circumcircle, orthocenter related

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Source: 2019 Argentina IberoAmerican Training List p15

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#1 h Aug 22, 2021, 10:54 AM • 1 Y

Y by mathmax12

Let $ABC$ be an acute triangle with $AB \ne AC$ and $H$ its orthocenter. Let $D$ and $E$ be the intersections of $BH$ and $CH$ with $AC$ and $AB$ respectively, and $P$ the foot of the perpendicular from $A$ on $DE$ . The circumcircle of $BPC$ intersects $DE$ at a point $Q \ne P$ . Show that lines $AP$ and $QH$ intersect at a point on the circumcircle of $ABC$ .

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#2 h Aug 22, 2021, 12:57 PM • 1 Y

Y by mathmax12

Let $DE$ intersects $BC$ at $W$ . Since $\overline{WP} \cdot \overline{WQ} = \overline{WB} \cdot \overline{WC} = \overline{WD} \cdot \overline{WE},$ we have $(WP, ED) = (ED, Q)$ . Then $\dfrac{\overline{EW}}{\overline{EP}} : \dfrac{\overline{DW}}{\overline{DP}} = \dfrac{\overline{QE}}{\overline{QD}}$ . So $\dfrac{QE}{QD} = \dfrac{EW}{EP} \cdot \dfrac{DP}{DW} = \dfrac{BE}{BD} \cdot \dfrac{CE}{CD} \cdot \dfrac{AD}{AE} \cdot \dfrac{HE}{HD} = \dfrac{HE^2}{HD^2}$ or $HQ$ passes through midpoint of $BC$

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