Ptolomy's Theorem

Let  ABCD be a cyclic quadrialteral. Ptolomy's Theorem says that the product of the diagonal lengths  equals the sum of the product of the lengths of opposite sides:

AC ^. BD = AB ^. CD + BC ^. AD

Proof:

Notice the angles ACB and ADB are congruent since they intercept the same arc (arc AB). Let P be the point on BD such that angle PCD is congruent to angles ACB and ADB.

In triangles ABC and DPC, angles BAC and BDC are congruent because they both share arc BC, and angles BCA and PCD are congruent by our choice of point P.  So these triangles are similar by AA and we can write the proportion:

Cross-multiplying gives us:

Also notice that triangle BPC is similar to triangle ADC by AA, so we also know:

and therefore

Putting this together, we have:

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