Mathematics

Mathematics

Problem

A,B,CA, B, C and DD are four successive points on a circle with centre OO. The chords ABAB and CDCD are equal, and so are chords ADAD and BCBC. The chords ACAC and BDBD intersect at point PP inside the circle. Carry out the following tasks: a. Prove that △APD\triangle APD is congruent to △BPC\triangle BPC. b. Hence show that ∠APD=∠BPC\angle APD = \angle BPC. c. Deduce that ∠AOB=∠COD\angle AOB = \angle COD.
Intuitive explanation

The key insight is recognizing that equal chords subtend equal arcs, which constrains the positions of these four points into a special symmetric configuration. Since AB=CDAB = CD and AD=BCAD = BC, the points form an isosceles trapezoid inscribed in the circle—a shape with a line of symmetry. Once you identify this symmetry, you can use it to determine relationships between the angles, arcs, and the position of intersection point PP without heavy calculation.

Intuitive explanation

The key insight is recognizing that equal chords subtend equal arcs, which constrains the positions of these four points into a special symmetric configuration. Since AB=CDAB = CD and AD=BCAD = BC, the points form an isosceles trapezoid inscribed in the circle—a shape with a line of symmetry. Once you identify this symmetry, you can use it to determine relationships between the angles, arcs, and the position of intersection point PP without heavy calculation.

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