Two chords AB and CD of a circle intersect at a point P inside the circle. Prove that AP × PB = CP × PD. State the theorem that justifies your answer to part (2).

Step 1: Identify given information and goal

We are given two chords, AB and CD, that intersect at a point P inside a circle. Our goal is to prove the relationship between the segments formed by this intersection: the product of the segments of one chord equals the product of the segments of the other chord.

Step 2: Construct triangles and identify similar angles

To establish a relationship between the segments, we can form triangles by joining AC and DB. By doing so, we can identify angles that are equal. Angles subtended by the same arc in a circle are equal, so $\angle PAC = \angle PDB$ (angles subtended by arc CB). Also, $\angle APC$ and $\angle DPB$ are vertically opposite angles, which means they are equal.

Step 3: Prove similarity of triangles

Since we have identified two pairs of equal angles ($\angle PAC = \angle PDB$ and $\angle APC = \angle DPB$), we can conclude that triangle PAC is similar to triangle PDB by the Angle-Angle (AA) similarity criterion.

Step 4: Use proportionality of sides to prove the relationship

Because triangles PAC and PDB are similar, their corresponding sides are proportional. This means that the ratio of AP to DP is equal to the ratio of CP to PB. By cross-multiplication, we arrive at the desired result: $AP \times PB = CP \times PD$.

Step 5: State the theorem

The theorem that justifies this result is known as the Theorem of Intersecting Chords. It states that if two chords intersect inside a circle, then the product of the segments of one chord is equal to the product of the segments of the other chord.

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