The inscribed angle (∠APB) looking at arc AB always stays the same size, no matter where point P sits on the circle. And that size is always exactly half of the central angle (∠AOB). Check it for yourself with the sliders.
The inscribed angle theorem: the inscribed angle on a given arc is always constant, and it equals half the central angle on that same arc. In particular, the inscribed angle on a diameter (semicircle) is always 90° — this is also known as "Thales' theorem."
The angle stays the same no matter where point P sits on the circle because the central angle is always fixed, and the relationship "inscribed angle = half the central angle" holds regardless of where P is. Once arc AB is fixed, the inscribed angle's size is automatically determined too.
Move point P around the circle with the slider and check whether angle ∠APB really does stay the same. Change the size of arc AB, and you'll see the inscribed angle and central angle change together.