Question 126
Given is an acute triangle $\triangle A B C$ with heights $\overline{A U}$, $\overline{B V}$, $\overline{C W}$ and height intersection $H$. $X$ is on $\overline{A U}$, $Y$ on $\overline{B V}$ and $Z$ on $\overline{C W}$. $X$, $Y$ and $Z$ are all different from $H$. Show
a. If $X$, $Y$, $Z$ and $H$ lie on a circle, applies
$$
[A B C]=[A B Z]+[A Y C]+[X B C]
$$
where $[P Q R]$ denotes the area of triangle $\triangle P Q R$.
b. The converse of (a) also applies.
a. If $X$, $Y$, $Z$ and $H$ lie on a circle, applies
$$
[A B C]=[A B Z]+[A Y C]+[X B C]
$$
where $[P Q R]$ denotes the area of triangle $\triangle P Q R$.
b. The converse of (a) also applies.