Properties of the orthic triangle

Question

In the following configuration, $PQR$ is the orthic triangle of $ABC$:

I have to prove (or find) different things:

1. The sides of the orthic triangle are antiparallel to the sides of $ABC$
2. The orthocenter of $ABC$ is the incenter of $PQR$
3. Which points are $A,B,C$ with respect to $PQR$?

I would be glad to receive some hints.

Answer 1

In order to prove 1., notice that $APQC$ is a cyclic quadrilateral and so on.
To prove 2., perform some angle chasing.
About 3.: if $H$ (the orthocenter of $ABC$) is the incenter of $PQR$, it follows that $A,B,C$ are the excenters of $PQR$. It is interesting to point out that 2. implies that the orthic triangle is the cevian triangle with the shortest perimeter.

Answer 2

2. Quadrilaterals APHR, PBQH, HQCR are cyclic. Triangle BQA is similar to triangle BPC. So angle PAH=PRH=HCQ=HRQ. Hence BR is Angle Bisector of angle PRQ and so on.Hence proved.
3. A, B, C are excentres of triangle PQR.