$2$. Let $ABC$ be an acute-angled triangle, and let $D$,$E$ be the feet of the perpendiculars from $A$, $B$ to $BC$, $CA$ respectively. Let $P$ be the point where the line $AD$ meets the semicircle constructed outwardly on $BC$, and $Q$ be the point where the line $BE$ meets the semicircle constructed outwardly on $AC$. Prove that $CP = CQ$.
I've haven't made much progress at all. Can anyone give a complete proof. Thanks in advance.
Relative to an origin O let the position of A be $\underline a$, and so on.
From the given information we have $$(\underline p-\underline a).(\underline b-\underline c)=0$$ $$(\underline b-\underline q).(\underline c-\underline a)=0$$
The angle in a semicircle is a right angle also gives us $$(\underline p-\underline c).(\underline b-\underline p)=0$$ $$(\underline q-\underline c).(\underline a-\underline q)=0$$
We need to show that $$(\underline c-\underline p).(\underline c-\underline p)=(\underline c-\underline q).(\underline c-\underline q)$$ which is equivalent to showing $$\underline p.\underline p-\underline q.\underline q=2(\underline c.\underline p-\underline c.\underline q)$$
To get this result, expand the third and fourth equations and subtract them. This gives $$\underline p.\underline p-\underline q.\underline q=\underline c.\underline p-\underline c.\underline q+[\underline p.\underline b-\underline c.\underline b+\underline a.\underline c-\underline a.\underline q]$$
However, if we expand the first two equations and subtract them we also get, after rearrangement, $$\underline p.\underline b-\underline c.\underline b+\underline a.\underline c-\underline a.\underline q=\underline c.\underline p-\underline c.\underline q$$ So the required result follows immediately
$\triangle BPC$ is a right triangle, and $\triangle PDC \sim \triangle BPC$, so
$$\frac{|\overline{CD}|}{|\overline{PC}|} = \frac{|\overline{PC}|}{|\overline{BC}|}\qquad \to\qquad |\overline{PC}|^2 = |\overline{BC}|\;|\overline{CD}|=|\overline{BC}|\;|\overline{AC}|\;\cos C \quad (\star)$$
Since the right-hand side of $(\star)$ is symmetric in $A$ and $B$, we'd get the same result for the point $Q$ constructed via the perpendicular from $B$ to $\overline{AC}$. Therefore, $|\overline{PC}| = |\overline{QC}|$. $\square$
Note: There's nothing special about $P$ and $Q$ being on the "outer" semi-circles. The circle about $C$ through $P$ (or $Q$) passes through all four points where the altitudes from $A$ and $B$ meet the circles having the opposite edges as diameters.
