The given information was insufficient to show that the conjecture is true. For that reason, we used the above graph to show it. To do so, we made some observations. Looking at the diagram, we saw that ∠ APB and ∠ CPD are vertical angles. Therefore, by the Vertical Angles Theorem, we concluded that they are congruent.
∠ APB ≅ ∠ CPD
However, this observation alone was not useful. That is why we measured ∠ BAD and ∠ DCB.
Both ∠ BAD and ∠ DCB have the same approximate measure. Therefore, we found that these angles are also congruent.
∠ BAD ≅ ∠ DCB
By the Angle-Angle Similarity Theorem, we concluded that △ APB and △ CPD are similar.
△ APB ~ △ CPD
Therefore, we could write a proportion between the corresponding side lengths of the triangles.
AP/CP=BP/DP
Then, by the Means Switching Property of proportions, we had APBP= CPDP. As a result, we justified the given conjecture, APBP= CPDP, based on our observations.
