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this is a problem of regional math olympiad 2015

Disclose the first approach

Use the Power of Point theorem to prove that $PA\times PD=PE^2=PB\times PC$

asif e elahi wrote:Use the Power of Point theorem to prove that $PA\times PD=PE^2=PB\times PC$

$PA \times PD = 12 \times (PA + AB + BC + CD) = 12 \times (12 + AB + BC + 2AB)$
$ = 36AB + 12BC + 144...(i)$
$PB \times PC = (PA+AB) \times (PA + AB + BC) = (12+AB) \times (12 + AB + BC)$
$ = AB^2 + 24AB + 12AB + AB \times BC ...(ii)$
$(i) = (ii) \rightarrow 36AB + 12BC + 144 = AB^2 + 24AB + 12AB + AB \times BC$
$ \rightarrow 12AB = AB(AB+BC)$
Thus,$AB+BC = 12$
And we get , $PC = PA + AB + BC = 24$