The chords AC and BD are drawn in the circle so that they intersect at point P.

The chords AC and BD are drawn in the circle so that they intersect at point P. Prove that the angle APB is equal to the half-sum of the angular values of the arcs AB and CD.

Let’s connect points B and C.

The inscribed angle CBD rests on the arc CD, which means that it is equal to half of its degree measure.

The inscribed angle ACB rests on the arc AB, and therefore is equal to half of its degree measure.

In the ВСP triangle, the ВРС angle = (180 – PBC – PСВ).

The APB angle is adjacent to the BPC angle, then the APB angle = (180 – BPC) = (180 – 180 + PBC + PCB) = (PBC + PCB).

Then the angle ABP = (⌒AB + ⌒СD) / 2, as required.